A quick-fire brace just after the break secured the points for the Hammers as Allardyce's side made it back-to-back Premier League victories for the first time this season.
West Ham squandered a number of first-half opportunities but their 31-year-old captain flicked them ahead just 18 seconds into the second-half before seizing upon a Fabian Delph error to double the lead.
The win sees Allardyce's men move two points clear of the bottom three and he says the result was deserved due to the number of chances they created.
"It was really sweet for Kevin to score two goals just after half time because I was bitterly disappointed we weren't winning at half time," he told Sky Sports.
"We could've really put the game out of sight of Aston Villa had we done better in front of goal or in the final third the amount of times we got in behind their defence and in wide positions, getting corners and crosses into the box.
"Not finding the net was something I was hoping we weren't going to regret today and Kevin started the second half with two outstanding finishes and he's the reason we're sat here with three points, a well-deserved three points.
"Then there was some outstanding defending after that when Villa threw three up the middle and we put our three centre halves in there and we just defended correctly. All the long balls into the box from Villa's display were masterfully dealt with."
Nolan's double took his tally for the campaign up to six and subsequently made him the Hammers' top scorer, much to the delight of Allardyce.
"He's up there again," he said. "He was leading goalscorer last season and he's pulling away from Carlton (Cole) with five now which is nice to see.
"Our goalscorers have started scoring on a regular basis. It really has been a missing link throughout the whole of the season never mind the injuries to the defenders in December.
"Aston Villa have been going pretty well themselves just recently and been scoring quite often so it's a terrific performance and a terrific result for us."
Speaking of an injury picked up by Nolan, Allardyce said: "Kevin came off with only a slight groin so I don't think it's going to be too much of a problem and even more enjoyable, we kept 11 on the field today."
The result also meant that West Ham have now kept the most Premier League clean sheets so far this season.
Allardyce said: "Considering the position we are in the league, we'd normally have 12 clean sheets and we'd be sitting in the top-half generally.
"But our season has been not scoring chances created and if we'd have done that better, we wouldn't have been in such a desperate position through December into January but we're beginning to pull away from that position."