After a goalless first half, Kevin Nolan scored twice in two minutes just after the restart to hand the east London club a vital win in their fight to avoid relegation.
West Ham, who were without the suspended Andy Carroll, moved up to 15th place in the Premier League table and two points above the drop zone.
And Allardyce told Sky Sports: “It was really sweet for Kevin to score two goals just after half-time because I was bitterly disappointed that we weren’t winning at half-time. We could really put the game out of sight had we done better in the final third.
“Kevin started the second half with two outstanding finishes – he’s the reason we’re sat here with three points and a well-deserved three points.
“There was some outstanding defending after that. We just defended correctly – all the long balls that were coming in the box from Villas’ play were masterfully dealt with.
“It’s a terrific performance and result for us, and even more enjoyable was that we kept 11 players on the field today!”
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