The Gunners headed to Upton Park looking to produce the required response following a disappointing 2-1 home defeat by Barclays Premier League leaders Chelsea, after having beaten Olympiacos in their midweek European tie.
Arsenal dominated the early stages, but were made to pay for not making the most of possession when Mohamed Diame put the Irons in front with a well-taken first goal for the club on 21 minutes.
However, the Gunners then dug deep to level before the interval through Olivier Giroud, his first in the Premier League, with substitute Theo Walcott (77) adding a second on the break and the industrious Santi Cazorla cracking in a 25-yard effort to wrap things up with seven minutes left.
Wenger's men will now head into the international break confident they can build a sustained attack on the top four.
"We started with a lot of questions raised about the team. We have now played seven games and the belief in the squad is that we have a chance," he said.
"Of course we now need to do it week after week. The spirit and quality are there. It is about how consistent we are now."
Wenger added: "It was important for us to respond well and after our defeat against Chelsea, we won against Olympiacos on Wednesday night.
"We had a difficult game three days later and we knew a draw here was not good enough for us, so we went for it and gave absolutely everything to win the game."
Giroud, a £12million signing from Montpellier, was recalled to the side on Saturday, and produced a determined display in a central striker role, converting Lukas Podolski's cross at the near post.
Wenger said: "He was good, had the physical presence and scored the goal. That will help him a lot to adapt very quickly."
Arsenal left-back Kieran Gibbs went off in the second half with a thigh problem, which could impact on his participation for England.
West Ham had certainly made a game of Saturday's contest, and on another afternoon could well have come away with a point, as Kevin Nolan and Andy Carroll both missed the target when well placed.
Midfielder Diame was, though, earlier somewhat fortunate to escape a second yellow card for a poor challenge on Mikel Arteta just minutes after being cautioned by referee Phil Dowd when he celebrated wildly in front of the home crowd after opening his account in some style.
England striker Carroll, meanwhile, showed few signs of being hampered by the hamstring problem which sidelined him for more a month as he completed 90 minutes ahead of the World Cup qualifiers.
Hammers boss Sam Allardyce said: "We didn't want to play him for 90 minutes because he has only just come back from that long spell injured and hasn't played that much but unfortunately our injury plight continued as the game progressed."
Allardyce - who expects to be without forward Ricardo Vaz Te for a couple of months with a suspected dislocated shoulder - feels his newly-promoted side can take confidence from their start, which has returned just two defeats from seven league games.
"There is cause for optimism because we have started the season very well," he said.
"It was a disappointing result, but not a disappointing performance.
"However, we can't linger on this disappointment. We have to get back to winning ways in two weeks' time.
"That will be the big test for us now - can we bounce back after a defeat and can we beat Southampton and make sure we get three points on the board again."