Pressure had been mounting on the Frenchman following his side's defeat in their opening Premier League game against Aston Villa, along with their struggles in the transfer market.
Wenger told the club's official website: "I feel relieved. Somebody told me yesterday that I've made 956 games for Arsenal. That means I care about what's happening at this club.
"When we had the result last week, I was of course deeply disappointed. I'm not on a personal trip any more. I'm about doing the things well for the club I love so I'm happy that we are back on track, that's all."
Arsenal face Fenerbahce next week in their Champions League qualifier before they take on Tottenham Hotspur in the north London derby next Sunday.