The result means Thomas Tuchel's side finish fourth in the league with 67 points after Leicester City lost to Tottenham Hotspur and Liverpool beat Crystal Palace, and will not need to rely on winning next weekend's Champions League final against Manchester City to return to the competition.
Despite Chelsea dominating most of the first half, Villa took the lead from a corner kick just before the break when former Chelsea man Bertrand Traore punched home a looping shot off the underside of the crossbar after being picked out by a smart rolled cross in the middle of the box.
Chelsea keeper Edouard Mendy was injured on the play after crashing into the post in his desperate attempt to make a save, and was eventually removed at half-time for Blues' backup Kepa Arrizabalaga.
Kepa was called on almost immediately when Villa were awarded a penalty in the 51st minute for a Jorginho clip on Traore. But the Spain keeper guessed wrong on Anwar El Ghazi's spot kick, seemingly leaving Chelsea a mountain to climb to finish in the top four.
Azpilicueta was sent off in the 90th minute after a violent clash with Jack Grealish, and Chelsea finished the match with 10 men.
After the match, Tuchel said the team will monitor Mendy's fitness ahead of the Champions League final.
"It is easy to handle [but] he fell into the frame of the net. He feels a big pain in his ribs," Chelsea manager Thomas Tuchel said.
"We will have an update tomorrow with what's going on. We will see if it is possible and if it is possible, we will push to do everything so he is in goal on Saturday."
Tuchel added that it was unsatisfactory way for Chelsea to scramble into fourth place.
"We are lucky that Tottenham did the job for us and finished the job for us today," said Tuchel, whose unbeaten away record since joining Chelsea in January came to an end.
"We came to win and to put any doubts to the side but it was not possible. That it was possible to end in the top four even with a defeat was because of the team's hard work and consistently good results.
"We need to adjust details and we need to be spot on on Saturday as we want a positive outcome."