The Saints have now surrendered 13 points from winning positions this season, and they were the masters of their own downfall after gifting Swansea an equaliser they hardly looked like getting.
Swansea boss Michael Laudrup satisfied by Saints point
Dyer, a graduate of Southampton's academy, took the chance well and gave Swansea their first point on the road since their
5-0 win at QPR
on the opening day of the season.
Southampton came into the match knowing that only once has a team not been relegated after losing eight of their first 10 games. The tension surrounding the club's predicament manifested itself in the stands, resulting in a subdued atmosphere at St Mary's.
The Saints certainly played the more direct football and had early sights at goal through Adam Lallana and Gaston Ramirez that failed to trouble Gerhard Tremmel in the Swansea goal.
Centre-back Jose Fonte was also denied at the far post by a lunging block from Swansea's 19-year-old left-back Ben Davies.
His opposite number, 17-year-old Luke Shaw, became the youngest Southampton player to make a Premier League start and acquitted himself well against Swansea's record signing Pablo Hernandez.
Most of Swansea's limited attacks veered towards Shaw's flank. The most significant opening of the first half released right-back Angel Rangel, but his low cross from inside the area was cleared at the front post with Michu unmarked in front of goal.
John MotsonBBC Match of the Day commentator
"Southampton manager Nigel Adkins didn't get the result he needed and the speculation over his position will intensify. At this stage of the season, a home draw is not enough, as I'm sure the Southampton hierarchy will agree."
The striker stood alone in the tunnel for several minutes at half-time, an indication of the visitors' growing frustration at the lack of any cutting edge or spark.
Home pressure finally began to grow after the break and Sung-Yueng Ki, who forced the only save from Southampton keeper Paulo Gazzaniga in the first half, made a crucial intervention to deny Lambert.
Garry Monk, making his 250th appearance for Swansea at the club that launched his career, then threw himself to block a Ramirez shot.
But the pressure finally told as Schneiderlin scored to release a wave of relief around St Mary's.
Lambert had won his fair share of headers and when he rose unchallenged at the far post to head across the face of goal, Schneiderlin showed great agility to chest the ball and then head past the advancing Tremmel.
Nigel Adkins hails good performances following draw against Swansea
The lead lasted all of nine minutes, though. Gazzaniga's pass put centre-back Maya Yoshida under unnecessary pressure and his poor first touch allowed Dyer to pounce.
The winger, a 56th-minute substitute, drove into the area and his low strike across the face of goal went in off the post.
Lambert twice threatened with headers but a sense of nervousness grew among the home defence and only a desperate lunge from Fox denied Shechter a tap-in.
Full Time The referee blows for the end of the match.
Corner taken left-footed by Daniel Fox, save by Gerhard Tremmel.
Unfair challenge on Ashley Williams by Rickie Lambert results in a free kick. Free kick taken by Gerhard Tremmel.
Free kick taken by Daniel Fox.
Booking Booking for Kemy Agustien.
Morgan Schneiderlin fouled by Kemy Agustien, the ref awards a free kick.
Miguel Michu sends in a cross, clearance by Maya Yoshida.
Pablo Hernandez delivers the ball.
Substitution (Swansea) makes a substitution, with Kemy Agustien coming on for Jonathan de Guzman.
Centre by Daniel Fox, Ashley Williams makes a clearance.
Free kick awarded for an unfair challenge on Garry Monk by Jose Fonte. Gerhard Tremmel restarts play with the free kick.
Inswinging corner taken by Gaston Ramirez.
Nathan Dyer is penalised for handball and concedes a free kick. Daniel Fox takes the indirect free kick.
Gaston Ramirez takes a inswinging corner.
Unfair challenge on Itay Shechter by Maya Yoshida results in a free kick. Sung-Yeung Ki takes the direct free kick.
The ball is swung over by Nathaniel Clyne, Rickie Lambert takes a shot. Save made by Gerhard Tremmel.
Effort on goal by Jack Cork from just inside the box goes harmlessly over the target.
Nathan Dyer concedes a free kick for a foul on Adam Lallana. Daniel Fox takes the direct free kick.
Corner taken left-footed by Gaston Ramirez, Rickie Lambert has a headed effort at goal from close in which goes wide left of the target.
Shot from 20 yards by Jason Puncheon. Garry Monk gets a block in.
Adam Lallana crosses the ball, Angel Rangel manages to make a clearance.
Itay Shechter is caught offside. Jose Fonte takes the direct free kick.
The ball is sent over by Daniel Fox, clearance by Ben Davies.
Substitution Daniel Fox on for Luke Shaw.
72:13 Goal scored
Goal - Nathan Dyer - Southamp'n 1 - 1 Swansea Nathan Dyer finds the net with a goal from the edge of the penalty area to the bottom left corner of the goal. Southampton 1-1 Swansea.
Nathaniel Clyne crosses the ball, clearance by Ashley Williams.
Itay Shechter is penalised for handball and concedes a free kick. Paulo Gazzaniga takes the indirect free kick.
Foul by Itay Shechter on Maya Yoshida, free kick awarded. Direct free kick taken by Paulo Gazzaniga.
Adam Lallana crosses the ball, Jonathan de Guzman makes a clearance.
Nathan Dyer challenges Adam Lallana unfairly and gives away a free kick. Paulo Gazzaniga restarts play with the free kick.
Itay Shechter is flagged offside by the assistant referee. Paulo Gazzaniga restarts play with the free kick.
Substitution (Swansea) makes a substitution, with Itay Shechter coming on for Leon Britton.
Rickie Lambert provided the assist for the goal.
63:51 Goal scored
Goal - Morgan Schneiderlin - Southamp'n 1 - 0 Swansea Morgan Schneiderlin grabs a headed goal from close in. Southampton 1-0 Swansea.
Booking Booking for Adam Lallana.
The ball is swung over by Luke Shaw, comfortable save by Gerhard Tremmel.
Gaston Ramirez takes a shot. Blocked by Garry Monk.
Morgan Schneiderlin concedes a free kick for a foul on Leon Britton. Direct free kick taken by Leon Britton.
Corner taken left-footed by Gaston Ramirez, clearance made by Miguel Michu.
Substitution Nathan Dyer on for Wayne Routledge.
Shot on goal by Wayne Routledge from just inside the area clears the crossbar.
Jack Cork gives away a free kick for an unfair challenge on Leon Britton. Garry Monk takes the free kick.
Adam Lallana delivers the ball, Ashley Williams makes a clearance.
Corner taken short by Jonathan de Guzman. The ball is crossed by Pablo Hernandez, clearance made by Jack Cork.
Angel Rangel has an effort at goal from just outside the box which goes wide of the right-hand upright.
Free kick awarded for a foul by Gaston Ramirez on Garry Monk. Direct free kick taken by Gerhard Tremmel.
Centre by Jose Fonte, clearance by Garry Monk.
Unfair challenge on Adam Lallana by Angel Rangel results in a free kick. Adam Lallana crosses the ball from the free kick right-footed from left wing, save made by Gerhard Tremmel.
The referee blows for offside against Gaston Ramirez. Free kick taken by Gerhard Tremmel.
The referee blows his whistle to start the second half.
45:00 +1:04 Half time
Half Time The referee blows for half time.
Foul by Rickie Lambert on Ashley Williams, free kick awarded. Direct free kick taken by Ashley Williams.
Ben Davies takes the inswinging corner, Garry Monk takes a shot. Save by Paulo Gazzaniga.
Outswinging corner taken right-footed by Jonathan de Guzman from the right by-line, Unfair challenge on Jose Fonte by Ashley Williams results in a free kick. Paulo Gazzaniga takes the direct free kick.
Shot on goal by Adam Lallana from just inside the area clears the crossbar.
Inswinging corner taken right-footed by Adam Lallana from the left by-line to the near post, clearance by Sung-Yeung Ki.
Free kick awarded for an unfair challenge on Adam Lallana by Wayne Routledge. Paulo Gazzaniga restarts play with the free kick.
The ball is crossed by Pablo Hernandez, clearance by Jose Fonte.
Free kick awarded for a foul by Rickie Lambert on Ben Davies. Gerhard Tremmel takes the free kick.
The free kick is delivered left-footed by Gaston Ramirez from left channel, Ben Davies manages to make a clearance.
Booking Booking for Ashley Williams for unsporting behaviour.
Free kick awarded for a foul by Ashley Williams on Gaston Ramirez.
Garry Monk takes the direct free kick.
Booking Booking for Morgan Schneiderlin for unsporting behaviour.
Free kick awarded for an unfair challenge on Leon Britton by Morgan Schneiderlin.
Inswinging corner taken from the right by-line by Gaston Ramirez.
Inswinging corner taken from the left by-line by Adam Lallana, Ashley Williams makes a clearance.
Luke Shaw produces a cross, Pablo Hernandez gets a block in.
Morgan Schneiderlin fouled by Wayne Routledge, the ref awards a free kick. Direct free kick taken by Paulo Gazzaniga.
Gaston Ramirez has an effort at goal from just outside the area which goes wide of the left-hand post.
Garry Monk concedes a free kick for a foul on Rickie Lambert. Jason Puncheon restarts play with the free kick.
Outswinging corner taken by Jonathan de Guzman, Rickie Lambert makes a clearance.
Sung-Yeung Ki takes a shot. Paulo Gazzaniga makes a brilliant save.
Corner from right by-line taken by Jonathan de Guzman.
Pablo Hernandez produces a cross.
Foul by Maya Yoshida on Miguel Michu, free kick awarded. Free kick taken by Leon Britton.
Shot from just outside the penalty box by Gaston Ramirez misses to the left of the target.
Shot by Adam Lallana from 20 yards. Save by Gerhard Tremmel.
Jonathan de Guzman produces a cross, save by Paulo Gazzaniga.
Outswinging corner taken from the right by-line by Jonathan de Guzman, clearance made by Jack Cork.
The referee gives a free kick against Jack Cork for handball. Free kick taken by Sung-Yeung Ki.
Effort on goal by Morgan Schneiderlin from just outside the penalty area goes harmlessly over the target.
Gaston Ramirez takes a inswinging corner from the right by-line to the near post, clearance made by Angel Rangel.
Inswinging corner taken from the left by-line by Adam Lallana, clearance made by Ashley Williams.
Jason Puncheon produces a cross, clearance by Angel Rangel.
Unfair challenge on Miguel Michu by Jason Puncheon results in a free kick. Direct free kick taken by Miguel Michu.
A cross is delivered by Angel Rangel.
The ref blows the whistle to begin the match.
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