Luke Moore grabbed a late goal to earn Villa a fully deserved draw in a thrilling game against leaders Chelsea.
Robben's strike was Chelsea 50th Premiership goal this season
Moore diverted Mark Delaney's cross over the line with 13 minutes left as home pressure finally told.
Arjen Robben gave Chelsea the lead when he reacted first to a William Gallas cross to fire in from 12 yards.
But Villa poured forward and substitute Moore got the better of Glen Johnson to deny Chelsea in a game which saw both sides denied obvious penalties.
While Chelsea failed to win in the Premiership for only the fourth time this season, they still managed to strengthen their dominant position at the top of the table.
Manchester United's defeat at Blackburn and Liverpool's home draw with Birmingham meant the Blues moved 15 points clear of the pack.
Villa had the best early chance when Gareth Barry blazed wildly over the bar after Peter Cech flapped at a James Milner cross.
The home side were made to rue that miss within seconds when Robben blasted Chelsea into the lead.
Gallas got free on the left and the Dutch winger reacted quickly to muscle Joe Cole out of the way and fire home his fifth goal of the season.
The goal forced Villa to go for broke and meant Chelsea could rely on the roving Robben and Joe Cole on the break.
Robben in particular was causing all sorts of problems and greedily tried to shoot at goal when all he needed to do was tee up Cole
But Villa were also looking lively in attack and playing a full part in an open encounter.
Gavin McCann saw a dipping effort go just wide and the guile of Milan Baros and Kevin Phillips, back following two months out with an ankle injury, was giving the Chelsea rearguard plenty to think about.
First the home side had an excellent shout for a penalty waved away by referee Rob Styles when the excellent Gareth Barry turned sharply in the box and John Terry blocked his run on goal with his arm.
Then a fine Phillips header was superbly tipped round the post by Cech while Barry also fizzed a shot just wide after a well-worked corner.
The frenetic pace continued after the break with Villa doing all the pressing and Chelsea relying on counter-attack football.
An almighty scramble saw a Phillips shot blocked by Gallas and Terry threw himself in front of of another Villa shot on goal.
Villa's pressure finally told when Delaney jinked his way past two defenders on the right flank and his low cross was bundled in by Moore - his second goal against the champions this season.
There was still time for Chelsea to have a strong shout for a penalty of their own when Jlloyd Samuel appeared to bundle over Robben.
But Villa held on to deny the Blues a sixth consecutive Premiership away win.
Aston Villa boss David O'Leary:
"The draw was the least we deserved. We played really well and took the game to them.
"We had a strong start all and we did wrong in the first half was give a stupid goal away. But we were the better before the break.
"We should also have had a penalty but so should they. Full credit to the players because Chelsea are a great team."
Chelsea manager Jose Mourinho:
It was a very good game played with a lot of intensity. They played to get a result with a lot of self-belief from the first minute.
"But the referee was not at the level of the game and at the level of both teams. John Terry gave away a penalty and Robben won a penalty.
"But a penalty in 35 minutes and a penalty in 85 minutes are two different things. It's not an excuse for the result. Villa deserved a point."
Aston Villa: Sorensen, Hughes, Delaney, Mellberg, Samuel, Milner, McCann, Davis, Barry, Baros (Angel 75), Phillips (Moore 75).
Subs Not Used: Taylor, Hendrie, Ridgewell.
Goals: Moore 77.
Chelsea: Cech, Johnson, Ricardo Carvalho, Terry, Gallas, Lampard, Makelele, Joe Cole (Duff 69), Crespo (Maniche 75), Robben, Gudjohnsen (Huth 86).
Subs Not Used: Pidgeley, Del Horno. Booked: Crespo, Johnson, Gallas.
Goals: Robben 14.
Ref: R Styles (Hampshire).