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.
All of the pre-match hype had been about what impact Reds forward Luis Suarez would have against the club which tried to sign the Uruguay forward for £40,000,001 in the summer.
In the end, though, the match was settled by a strike of the highest order from Welshman Ramsey, who netted his 10th goal of an impressive individual campaign.
Reds boss Brendan Rodgers said: "We are disappointed with the result, but sometimes when you are beaten by a better team, you have to accept that.
"It was a game they deserved to win. They had greater control, but we were always in the game because of our ability to create chances."
The outcome, however, could have been different but for a disallowed Liverpool goal in the first half which would have made it 1-1.
After being pushed over by Bacary Sagna, Suarez took a quick free-kick which released Daniel Sturridge who crossed for Jordan Henderson to tap into the net - only for referee Martin Atkinson to pull play back and book the Arsenal defender.
Rodgers was not impressed. The Northern Irishman said: "I got an explanation from the fourth official on it, and it is disappointing, to be honest.
"He said it was the next phase of the game, so he was given a yellow card, and it could only have been given at that point.
"I thought it was a poor decision because you have to let the game flow, but that was the reason he gave it. It was clever play (from us), and we got a goal from it."
Liverpool and Chelsea, who lost at Newcastle, are now both five points behind the Gunners.
Rodgers added: "There is a long way to go. We have made an excellent start and I can't ask any more from the players. They are pushing every game."
Defender Glen Johnson was missing through illness, which meant Jon Flanagan came into the side.
Rodgers said: "Half an hour before I named the team, Glen had to pull out.
"He had to go to hospital with an illness. We hope he will be out (of hospital) this evening."
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