Egyptian midfielder Trezeguet's first-half goal was enough to give Villa the win.
"It was unbelievable, I think it was deserved we were outstanding today," Villa midfielder Jack Grealish, who was superb throughout, told Sky Sports.
"It's been a tense day, all we can do is focus on ourselves. It's up to us what we do on the last day. It feels massive to be out of the relegation zone," he added.
Mike Arteta made six changes to Arsenal's lineup that had beat City in Saturday's FA Cup semifinal.
"Emotionally and physically it's been a very demanding week but I saw a team whio wanted to fight until the end. I'm not disappointed about the attitude -- I know where we are in certain areas and where we need to improve more. We will get that," Arteta said.
Tuesday's loss also seemingly ensures 10th place Arsenal can only reach the Europa League next season by winning the FA Cup final against Chelsea on Aug. 1 (stream the FA Cup final in the U.S. on ESPN+).
Arsenal, who can no longer finish than eighth, are also guaranteed to have their worst Premier League finish since 1995.
"It tells you the gap doesn't lie. We have to make that smaller and smaller," Arteta added.
Trezeguet was one of many signings Villa made after their promotion last season, and the £8.75 million he cost might prove to be the club's most important investment.
He scored both goals when Villa beat Crystal Palace in their last home game and delivered again.
When Conor Hourihane swung over a corner in the 27th minute the ball came out to the Egyptian stationed on the edge of the area and he set himself before hammering a shot past Arsenal keeper Emiliano Martinez.
It was a nervy night at an empty Villa Park with Keinan Davis wasting a glorious chance to make it 2-0 and minutes later Arsenal's Edward Nketiah headed against the inside of the post.
Grealish went close before halftime and while Arsenal had plenty of possession after the break they looked flat.
Davis could have settled Villa's nerves when Grealish played him through but his left-foot shot slid wide.
That miss almost came back to bite Villa when Nketiah's glancing header struck the inside of the post and rebounded back into the hands of Pepe Reina who kissed the ball.
"It's a big boost, that's for sure. We knew we had to win to catch Watford up. Now we've got the season in our own hands and that's all you can ask for," Villa manager Dean Smith said.
Villa, now in 17th place with 34 points, are ahead of Watford on goal difference going into Sunday's final round of matches. Second-bottom side AFC Bournemouth, who next play Everton, have 31 points and also remain alive.
Norwich City, with 21 points, have already been relegated.
Villa will fight for their fate against 16th place West Ham United while Arsenal will play Watford in their respective closing league matches.