The Gunners had seen questions raised over their character for a sustained title bid following successive home defeats by Borussia Dortmund in the Champions League and after they crashed out of the Capital One Cup against Chelsea on Tuesday night.
Santi Cazorla fired the Gunners ahead somewhat against the run of play on 19 minutes, lashing the ball home after his bullet header had come back off the post.
Wales midfielder Aaron Ramsey then made sure of all three points on the hour when he cracked in a sublime 25-yard half-volley as Wenger's side renewed belief ahead of next week's European trip to Dortmund and Premier League game at Manchester United.
"It was important to respond today, to convince people that we can win these big games, to win it in a convincing way is even more important," the Arsenal manager said.
"I am always confident, but of course there is a long way to go. Today was an important win for the team, for the club, and overall nobody would dispute that we deserved to win this game.
"It was vital to win, I said that before the game and I don't change my mind, because had we not won people would say 'yes, Arsenal at home they cannot win the big games'.
"It is done, in a convincing way. We had a good solidarity, a good focus, and I believe we have plenty of other big games coming up and this will help us deal well with them."
Wenger continued: "Tactically we dealt well with what Liverpool offered us today in the first half.
"We stopped them from playing and overall we are defensively a better team."
Arsenal were without England midfielder Jack Wilshere, who has been troubled by an ankle problem.
Wenger said: "I don't know if he'll be available on Wednesday (against Dortmund). We have to wait until Monday. It is not long term."
Defender Kieran Gibbs is also a doubt for the trip to Germany after coming off late in the second half.