The Gunners scored four goals inside 10 minutes after the break to inspire the rout and get back on track after a three-game winless streak in the Premier League.
Wilshere was pleased with their improved showing in the second half after the opening period ended 1-1 and has stressed that they cannot afford to drop points now as they push for UEFA Champions League football.
He told Sky Sports: "It was a tough first half for us. We knew what West Ham bring: they're a physical team and they get behind the ball and defend.
"They got a good goal so we had to come out and play and in the second half for 15 minutes we really upped our game.
"We know the football we can play and that was that.
"It was a good game for us and we're happy with the three points."
He added of the race for European football: "There's no point having games in hand if you're not going to win them. We have to keep fighting for the fourth place and we can't afford to drop any points now."