Scotland worldwide Matt Ritchie's third goal of the season added the Gunners' scalp to the one he collected from Manchester United in February after Ayoze Perez had cancelled out Alexandre Lacazette's opener.
Newcastle has only lost one of its last seven games, winning five of them.
Wenger said, "It's the story of the season". "We didn't put our chances to bed and made unbelievable mistakes".
Traditionally we are strong away from home. It doesn't make sense. "We came out of it with no points and overall we thought we should have got three points".
Matt Ritchie then netted his third goal of the season to secure Newcastle's win and take them to 41 points for the season.
The Emirates outfit will play Atletico Madrid in the Europa League semi finals for a chance to play Champions League football next season if they emerge champions.
Alexandre Lacazette opened the scoring for the Gunners, and it looked to be heading for a routine win for Arsene Wenger's men, with the club having claimed victory over the Magpies in the two side's last 10 outings. Just moments after he was left writhing on the turf following a crunching challenge from Kenedy, Lacazette exacted revenge.
Pierre-Emerick Aubameyang picked up the ball down the Arsenal left and after racing into the Newcastle penalty error, played a fine square ball for his strike partner, Alexandre Lacazette to score.
Newcastle levelled when Mustafi was beaten to a near-post cross for the second league game in a row - this time by Perez, who deftly guided DeAndre Yedlin's pacy cross past Petr Cech.
Arsenal: Cech; Chambers (Maitland-Niles, min. 78), Mustafi, Holding, Monreal; Elneny, Xhaka, Willock (Welbeck, min. 68); Iwobi (Nketiah, min. 86), Aubameyang, Lacazette.
His finish took Arsenal to a century of goals in all competitions this season, becoming the fourth Premier League club to achieve that feat after Manchester City, Liverpool and Spurs respectively.
Perhaps encouraged by an Arsenal team sheet showing six changes from the side which had drawn away to CSKA Moscow on Thursday, the Magpies started at a high tempo but soon found themselves forced back into their own half with Lacazette scuffing a fifth-minute shot wide after turning neatly on the edge of the box.