Mikel Arteta’s side took the lead through Declan Rice’s first half opener on his 100th appearance for the club.
But Dean Huijsen equalised in the second half and Evanilson completed the comeback to give Bournemouth their first ever away win over Arsenal.
Beaten 1-0 by PSG in the Champions League semi-final first leg on Tuesday, Arsenal will need a much-improved performance in the second leg in Paris on Wednesday.
The second-placed Gunners also still require six points to guarantee Champions League qualification for next season via a top five finish in the Premier League.
Arsenal finished as runners-up behind Manchester City for the last two seasons and the Champions League is their last hope of a trophy this term.
They last reached the Champions League final in 2006 and have never won the tournament.
“We should have put the game to bed. We defended really poorly, nowhere near the standards that we are used to,” Arteta said.
“We wanted to generate the momentum and a really positive vibe today and we haven’t.
“What we have created is frustration, rage and anger. Make sure we use that on Wednesday.”
Unai Emery’s side are back in the hunt to finish in the Premier League’s top five thanks to Youri Tielemans’ 12th-minute header at Villa Park.
Seventh-placed Villa extended their unbeaten run on home turf to 20 games in all competitions, lifting them level with sixth-placed Nottingham Forest and fifth-placed Chelsea on 60 points.
After bouncing back from their dismal FA Cup semi-final loss to Crystal Palace last weekend, Villa will hope their rivals falter over the weekend as the top five race heads for a dramatic climax.
“If we weren’t winning today, forget it completely,” Emery said.
“Today, we got more or less to be close to Europe. Wow, fantastic, brilliant. The best gift we can achieve after it is the Europa League or Champions League.
“We will fight for it, Europe, and hopefully we can fight for the Champions League.”
Southampton wasted a chance to avoid a share of the lowest points total in Premier League history as Jamie Vardy’s 199th Leicester goal inspired a 2-0 win at the King Power Stadium.
Simon Rusk’s bottom of the table team are trying to avoid finishing with just 11 points — which would match Derby’s low mark set in 2007-08.
But Southampton’s hopes of getting a 12th point quickly faded against fellow relegated side Leicester.
After announcing last week that he will leave the club at the end of the season, the 38-year-old Vardy bagged a farewell goal as he put second-bottom Leicester ahead in the 17th minute.
The game was delayed for 11 minutes when referee David Webb was forced off with a head injury after colliding with Leicester’s Jordan Ayew.
Vardy cheekily blew the referee’s whistle while he was being treated.
Leicester finally won for the first time since January as Ayew grabbed the second goal in the 44th minute.
Southampton have now lost 28 of their 35 league games and have three matches left to avoid sharing the low points record with Derby.
Relegated Ipswich fought back from two goals down to draw 2-2 in Everton’s penultimate match at Goodison Park.
Kieran McKenna’s third-bottom side trailed to Beto’s 26th-minute goal and Dwight McNeil’s effort nine minutes later.
But Julio Enciso’s 30-yard rocket reduced the deficit in the 41st minute and George Hirst equalised with 11 minutes left.
Everton’s emotional last match at Goodison before moving to their new Bramley-Moore Dock stadium next season comes against Southampton on May 18. (AFP)
