Speed, who won 85 caps for Wales, was found dead at his Cheshire home on Sunday morning.
A pre-match minute of silence at the Liberty Stadium spontaneously gave way to respectful applause and chants of "there's only one Gary Speed".
Substitute Leroy Lita had the game's best two chances for the hosts.
He fired wide with only Shay Given to beat with 20 minutes left, before the Villa goalkeeper saved well with his legs from Lita's close-range effort in the closing stages.
Match a tribute to Speed - Rodgers
But the game seemed of secondary importance on an emotional day in south Wales.
Swansea's Joe Allen, Neil Taylor and Ashley Williams - as well as Villa's James Collins - had represented Speed's Wales squad, while Given, who played alongside Speed during his spell at Newcastle, was in tears prior to kick-off.
Substitute Jermaine Jenas also knew Speed from his spell with the Magpies, and Swans midfielder Leon Britton was signed by Speed during his time as manager of Sheffield United.
There were questions as to whether the game would go ahead, but Swansea confirmed less than an hour before the scheduled start time that it would be played at 1330 GMT as planned.
Swansea have now kept six clean sheets this season, the joint-highest number in the division with Manchester United. Swansea's Michel Vorm has had more shut-outs than any other goalkeeper (six), one more than Fulham's Mark Schwarzer
When it did begin, the Swans started brightly, mounting plenty of pressure on the visitors' goal. Scott Sinclair and Joe Allen both had efforts from the edge of the area blocked by a stubborn Villa defence.
But the away side soon settled and they forged a succession of chances, Gabriel Agbonlahor firing straight at Swansea goalkeeper Michel Vorm from 15 yards out, Richard Dunne heading over from close range and Agbonlahor sending another effort from the edge of the box just wide.
Agbonlahor's pace down Villa's right continued to cause problems for the hosts, who lost defender Angel Rangel midway through the first half.
And the away side felt they could have been awarded a penalty just prior to the interval when Emile Heskey appeared to head the ball against the arm of Williams in the Swansea box from Charles N'Zogbia's cross.
In the opening minutes of the second period, Fabian Delph's sweeping left-foot pass picked out the advancing Alan Hutton on the right-hand side and the Villa defender brought it down well on his chest before firing a right-foot volley just over the top.
McLeish concerned for players
The home team were dealt a blow soon after when in-form striker Danny Graham was forced off through injury, to be replaced by Lita.
Agbonlahor continued to be Villa's stand-out man as the game wore on, cutting in from the left before sending a fierce right-footed drive towards the top corner, which Vorm unconvincingly tipped just over the crossbar.
And Villa were beginning to boss proceedings with N'Zogbia, returning to the starting line-up in place of the injured Stiliyan Petrov, enjoying some space inside the Swans' half, and Delph, in for Carlos Cueller, grabbing the initiative for the visitors in the middle of the park.
But the hosts gradually came back in to it and they had the better chances to take all three points.
Lita was fed through by Mark Gower with 20 minutes left, turning Dunne before placing his side-foot effort just wide of Given's left-hand post from inside the box.
Shortly after, the ever-threatening Sinclair received the ball on the left and jinked beautifully past Alan Hutton before his goalbound shot was blocked by Collins.
Sinclair then fed Lita inside the area with 10 minutes to go, and the substitute got away from Dunne before his close-range effort was well saved by Given.
Swansea manager Brendan Rodgers:
"I thought it was a massive respect to both teams in terms of the performance level they put on for the spectators.
"We wanted to play the game today for Gary Speed - he was such a good player and a great football man.
"Overall our performance level was excellent; just the final pass and the final shot to score the goal was missing but I was really proud of how we passed the ball and some of our movement."
Aston Villa manager Alex McLeish:
"It was devastating news for the whole of football and certainly for Gary Speed's family.
"We did try to use it as part of the motivation to say 'go and put on a spectacle for Speedo'.
"We played a very good Swansea team - they keep the ball well, they pass it well and I felt we only got penetrated in the last 10 or 15 minutes."
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