Ballymena salvaged a point thanks to Gary Thompson's goal three minutes into added time, despite playing with 10 men for all but four minutes of the game.
Allan Jenkins was sent-off for bundling down David Scullion inside the area and Fra McCaffrey scored the resulting penalty in the fifth minute.
Darren Boyce scored with a sublime 33rd-minute left-foot volley, but Aaron Hogg saved Ally Teggart's penalty.
Jordan Stewart poked home on 87 minutes but Thompson struck to secure a draw.
Ballymena began the day hoping to defend their 100% home record, while Eddie Patterson's Glens had yet to win away from home so far this season.
Matthew Tipton poked the ball wide in the early stages, before Jenkins was shown a red card for denying Scullion a clear goal-scoring opportunity a few yards out.
McCaffrey's penalty was well-taken and the Glens could have extended their lead but Aaron Stewart headed David Howland's header off the line and Niall Henderson was unable to convert another chance.
The Sky Blues equalised when Boyce found the bottom corner after getting on the end of Johnny Taylor's perfectly executed long ball out of defence.
Hogg then saved Teggart's penalty low to his right after Calum Birney was adjudged to have handled inside the area, the latter being fortunate to remain on the pitch after earlier receiving a yellow card.
After half-time, Tipton skewed a shot wide when well-placed and Scullion skied a left-foot attempt over the bar.
Referee Arnold Hunter disallowed a 'goal' when Curtis Allen found the net, heading in from a corner after a goal-kick looked to have been the correct decision.
Jordan Stewart struck the woodwork but poked the ball over the line three minutes from time after Dwayne Nelson had saved Allen's initial headed effort.
The visitors looked to have secured the points but Thompson rifled home deep into injury-time after Mark Surgenor's initial effort had been saved by Hogg.