Yossi Benayoun gave the Gunners an early lead when a Marton Fulop error allowed the Israeli to tap into an empty net in the fourth minute.
But the Londoners were pegged back when Shane Long fired home despite replays suggesting he was offside – and Graham Dorrans then put Roy Hodgson’s men ahead in the 15th minute.
However, goals from Andre Santos and Laurent Koscielny secured a vital win for Arsenal and a route into the group stages of Europe’s elite club competition.
“I think if you look at the season as a whole, we lost only two of the last 16 games and created a fantastic run,” said Wenger after the game.
“In the end we finished with 70 points, which is respectable, and we qualify for the Champions League for the 15th consecutive year. Of course we are very proud of that, especially this season having started where we started.
“We had problems getting over the line and you could see that again today.”
The result means Arsenal’s north London rivals Tottenham – who beat Fulham 2-0 – face an anxious wait to discover if they will be playing in next season’s Champions League, as Chelsea would qualify at Spurs’ expense if they beat Bayern Munich in next weekend’s final.
Wenger added: “I am very proud of this season because we were not only tested on the football front – which as a club we are used to – but on our mental solidity, unity and solidarity within.
“We were deeply tested, we did not show any weakness, kept united and in the end came back. It is a good lesson for everybody. At some stages everybody was wondering what we were doing.”
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news