Nolan scored twice in three minutes at the start of the second half as West Ham made it seven points from a possible nine, climbing out of the relegation zone as a result.
Allardyce admitted to a sense of relief after Villa failed to punish some wasteful finishing from his side prior to the interval.
"I was giving the lads a little bit of stick at half-time about not getting the lead after so much really good attacking play and stretching Villa’s defence on several occasions," he said.
"I thought we'd wasted our opportunity to take the lead on several occasions and we should have done our final bit a lot better than that.
"I said to the players, 'don't let that bite you in the backside today'.
"You don't expect them to go out and score two in five minutes, which Kevin did. We effectively finished Villa off at that stage.
"Kevin's quality at finishing comes to the front again. It's back-to-back wins for us, which at this stage in the season, where we were in the league, it's relieved a lot of pressure on us.
West Ham have now kept 12 clean sheets this season in the Premier League – more than any other side – and their latest shutout owed much to centre-back pairing James Tomkins and James Collins, as well as Winston Reid, who came on as a substitute in the second half.
"Professionally I thought we were outstanding and defensively, when they threw more men up front and started pumping high balls into the middle of the box, Winston Reid, 'Ginge' (Collins) and James Tomkins stood up to that aerial threat brilliantly," added Allardyce.
"We've got another clean sheet – 12 this season now – and things are looking a lot better. We've got 10 points from the last five games and hopefully we'll continue a winning run on Tuesday against Norwich."
