Bournemouth is their first victory in history at Arsenal, while Evanilson's tough goal gave Cherries a 2-1 victory in Saturday's Premier League.
Declan Rice won the Gunners' lead in the 34th minute, running to Martinødegaard's neat reverse pass and round goalkeeper Kepa Arrizabalaga before finishing his 100th Arsenal appearance.
The host had a chance to expand the lead, Bukayo Saka curl widened, but Dean Huijsen gave Bournemouth a well-deserved equalizer at No. 67, when he put a long throw from Antoine Semenyo to the bottom corner.
Eight minutes later, Evanilson tied home from the corner to complete the turnaround, after a lengthy var review determined that there was no certain evidence that he handled the ball.
"We achieved our goal and it felt like we had control of almost everything," Arsenal captain Ødegaard told Sky Sports. "But they scored, we lost control of the game...the second half was unacceptable."
Arsenal couldn't create another real opportunity and lost 21 points from the winning position of the season, which now depends on a 1-0 deficit in the second leg of the Champions League semi-finals of the Paris Saint-Germain champion on Wednesday.
Bournemouth's victory means they finished Arsenal's doubles for the first time and advanced to eighth, which may be enough for European football next season, with 53 points in 35 games.
AFC Bournemouth / AFC Bournemouth by Getty Images
Cherry's running is tough and visits Manchester City from Aston Villa next Saturday, but Evanilson is sure Bournemouth is eligible for Europe.
“It’s been a project submitted to me since I arrived,” he said. “I have always believed in the quality of fighting for these positions.”
Arsenal ranked second in the second place in champion Liverpool and ranked third in Manchester City with 67 points. Their remaining three games require six points to ensure the return of the top five next season and the Champions League.
Arsenal's only chance with Liverpool's crowned champions last weekend was the Champions League, but owner Mikel Arteta still calls it almost the first choice aspect.
PSG has been in the bag with Ligue 1, making 10 changes on Saturday in Strasbourg's 2-1 failure, but Arteta says he can't afford the same thing.
"They won the luxury," he told reporters. "We didn't, because we still have a lot to do in the Premier League, but we have four days and I'm sure we'll be fully recovered."
Rice proved Arsenal's greatest offensive threat in the first half, with two shots preventing him from returning home before defender Illia Zabarnyi performed well.
Evan Nelson then wasted a great opportunity, heading to Miloz Kerkez's beautiful ball in the 26th minute as Bournemouth posed a constant threat on the counter.
Ten minutes after the break, Saka approached Arsenal's lead, cutting the inside and shooting narrowly before Bournemouth snatched all threes with Quickfire Double.
Semenyo's first involvement was sloppy, and it was a bad feeling that cleared William Saliba, but the forward threw the final throw into the box to get Huijsen home.
Evanilson then made up for his earlier miss, stretching out Ødegaard on the back, turning Marcus Tavernier's flick towards past goalkeeper David Raya and wild Bournemouth fans.
The second goal seemed to remove Arsenal’s beliefs and by the end of the game, home fans were streaming towards exits, their ideas had moved to Paris on Wednesday.
This report uses Reuters information.