Wayne Routledge scored his first Premier League goal in his 116th appearance in the division two minutes into the second half to double their advantage, and Villa never looked likely to get back into the game.
Villa won 14 corners to Swansea's none, but that does not reflect a match which the Welsh side dominated at times - and they defended their goal stoutly when called upon.
Villa, who claimed a brilliant 3-1 win at Chelsea on Saturday, welcomed Darren Bent back into their starting XI. Bent, who missed the previous three games through injury, came on to notch the third of those goals at Stamford Bridge.
Captain Stiliyan Petrov also made it despite limping off soon after scoring the second goal in West London.
Left-back Warnock had a nightmare first half up against nippy right winger Dyer and put his side on the back foot early on when he under-hit a pass back to a retreating Richard Dunne. Dyer raced to the edge of the penalty area and struck a sweet shot into the far corner, although it took a deflection off the heel of Dunne.
Three minutes later Andrea Orlandi, making his first Premier League start for Swansea, drove forward and struck a low shot inches wide of the far post.
After a quarter of an hour Dyer again went close, his dipping shot from the right corner of the box whistling past the far top corner.
With the exception of the corners - with only one in the first half won cleanly by a Villa player, when James Collins's effort towards the top corner was claimed well by Michel Vorm - Villa created nothing, with Gabriel Agbonlahor shoved out on the right and Bent futilely jumping for everything in the middle.
Just before that Collins header, Agbonlahor went down looking for a penalty but referee Anthony Taylor was not having any of it: Neil Taylor had hold of his hand, but the contact did not warrant either the dive or a spot-kick.
Before the half-hour mark Dyer controlled brilliantly to put Warnock on his backside and cut back to Graham, whose shot was smothered by Brad Guzan.
Six minutes on Dyer led a counter-attack which saw Graham play in Orlandi down the left inside the box. When the ball came back to him, the striker's placed finish was just wide of the far post.
Villa had lost their three previous home games but they came against the illustrious opposition of Manchester United, Liverpool and Arsenal, and so their fans would not have been ready to throw in the towel at the break.
They might have been had they been aware that their side have not overturned a half-time deficit to win for three years, and lost 10 of their last 12 matches when behind at the interval.
And the chances of them breaking that run seemed unlikely at best when Routledge scored Swansea's second. The jet-heeled winger attacked from the left and pulled back to Graham, whose scuffed shot came back off the near post and Routledge was on hand to score from a tight angle past two defenders on the line.
The remainder of the second half was something of a non-event for the most part as Swansea, who had pressed up the pitch in the first half, dropped back to defend the incessant Villa crosses from within their box.
One humorous moment arrived after an hour when the tiny Dyer squared up to mammoth centre-half Dunne, who shoved him away dismissively.
The home fans gave vent to their frustration when Alex McLeish replaced Ireland, impressive against Chelsea but lacking in this encounter, with Marc Albrighton. Their chants of "You don't know what you're doing" were perhaps unfair given their win at Stamford Bridge only 48 hours earlier.
The tired legs began to show as Swansea went close late on through Graham - played in by Kemy Agustien - and Joe Allen, who almost rounded the brave Guzan after good work by substitute Scott Sinclair.
And Villa's afternoon was summed up four minutes into injury time when N'Zobgia, the taker of all their corners, struck a quite brilliant half-volley against the inside of the post from the edge of the box.
Swansea move above Villa on goal difference - by one goal - with the two-goal win. They stand 11th and 12th respectively.