West Ham have been in or around the relegation zone all season, and were level on points with their opponents heading into last night’s fixture.
But another three points and another clean sheet has seen Allardyce’s side leap to 10th position in the table.
Allardyce was made to sweat by his team, who waited until six minutes from time to break the deadlock.
Substitute Mohamed Diame crossed for James Collins to head home his first Premier League goal in six years, before Diame made it 2-0 in added time with the aid of a deflection.
“We are delighted in the end because it was a hell of a game Norwich gave us tonight,” Allardyce said.
“It was a really hard fought game but I think in the end our goalkeeper has kept us in the game and then the set pieces have put us on the right track.
“Obviously when you score that late on it’s difficult for the opposition to come back and when they throw everybody forward, Mo Diame was able to finish it off for us.”
“It was a great end to a great run of fixtures for us,” he added.
“We have a few days now where we can regroup, there is a lot of tired legs but not so many tired minds.
“We are in a great mood at the moment and this is a really big and hard-earned three points for us.”