The title-chasing Londoners had Willian sent off midway through the second half with the score at 0-0 and lost Ramires to another red card in the closing stages, with visiting manager Jose Mourinho sent to the stands for his reaction to referee Chris Foy's decision.
Man of the match Delph said: "I thought we were the better team from the start and we looked like we wanted it more. We attacked when we had the chance and we defended well."
The decisive goal arrived eight minutes from time. Delph won possession in midfield and passed to Marc Albrighton, who returned the ball for the former Leeds midfielder to back-heel superbly past Petr Cech.
Delph continued: "We've been scoring goals like that in training and it's nice for it to come off in a game. It was my best goal so far in the Premier League. We've been working hard since the last game. No-one gave us a chance. We needed to get a job done here, and luckily we've done it."
Delph missed the chance to double Villa's lead late on, and admitted: "I was a bit disappointed, to be honest. I thought I should have put it away. That'll haunt me a little bit but, at the end of the day, the three points was the important thing."
The win made it two on the bounce for Paul Lambert's Villa, who now sit 10th with 34 points, ending Chelsea's 14-match unbeaten run in the Premier League.
Chelsea's lead at the top was cut to six points after second- placed Manchester City's 2-0 win at Hull in the early match.
Mourinho decided not to comment on the three sendings off for his side - including his own dismissal to the stands - and instead tried to focus on Chelsea's performance.
He said: "It was a solid performance against a good team. Defensively, we were really slow and we had problems to cope with the danger Aston Villa had, but after that I prefer not to comment.
"If you comment you are in trouble. I prefer not to speak about the decisions. If I speak I am in big trouble."
Chelsea have two difficult matches coming up with their midweek home Champions League clash against Galatasary and next weekend's battle against Premier League rivals Arsenal.
The Portuguese admitted it will be difficult, but he reaffirmed he aims to keep quiet about today's goings on, adding: "We have to put it behind us, there is no other solution."