The Gunners, who had to win to guarantee Champions League football next season, got off to a good start at The Hawthorns when Yossi Benayoun put the visitors 1-0 up in the fourth minute.
But two quick-fire West Brom goals by Shane Long and Graham Dorrans handed Roy Hodgson’s men the initiative, before Andre Santos smashed in a leveller before the break.
Laurent Koscielny then turned home the eventual winner in the 54th minute to secure third place for the Gunners.
“It is good to sleep better at night,” Wenger told Arsenal Player. “It is a tricky position in fourth because you don’t want Chelsea to win the Champions League.
“And to be fourth is playing a qualifier, which every year is more difficult. Therefore just to be in the group stage straight away is fantastic.
“It is true that we had problems getting over the line but we were not the only ones. The players have given a lot, I must say.
“I looked at the players who were out and you see we had no [Mikel] Arteta, no [Bacary] Sagna and all the others who were out but we still managed to win away from home. That is down to remarkable character and I am very proud of the achievement of the team.
“I maybe take the biggest ever [satisfaction from this season] because at some stage of the season everybody would have said you were absolutely mad. We still kept our focus and worked hard together and in the end it is a good present to give [assistant manager] Pat Rice.”
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news