Nolan's close-range strike five minutes before the interval was enough for West Ham to make a successful return to the Barclays Premier League.
The goal came with a hint of controversy after Ricardo Vaz Te had been flagged off-side in the build-up, only for the linesman to change his mind.
Allardyce felt his side could have won by more - Carlton Cole had a penalty claim turned down and substitute Modibo Maiga saw a late shot cleared off the line.
But he was relieved that West Ham had made a strong start to the season.
Allardyce said: "Kevin Nolan is like David Platt because he ghosts into the box without anyone knowing how he got there.
"We might have won by more. It was a good performance with a clean sheet.
"It was a very good decision by the referee because the ball clearly came off the Villa player (when it was flicked on to Vaz Te) but refs don't always question their linesmen.
"We should have had a penalty when Carlton Cole was shoved to the ground.
"I was a bit nervous because you never quite know how it's going to go after pre-season. A good start is so important to a team such as us because it helps secure your Premier position as early as possible.
"It takes the pressure off the players and the fans, and it makes everyone enjoy their football."
Allardyce hailed the Hammers' defensive performance, which was marshalled by man of the match James Collins.
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The Wales centre-back was making his second debut for West Ham against his old team, after returning to Upton Park in the summer from Villa.
"It's been a very, very good day for us today [Saturday] because we won our first game of the season and kept a clean sheet as well," Allardyce said.
"To limit Darren Bent to three touches speaks volumes of our defensive performance and it seems we've hit the ground running."
Aston Villa players initially appealed against the decision to award the goal - but manager Paul Lambert accepted the right decision had been made.
Mark Noble's free kick glanced off the head of a Villa defender - not Collins as the linesman first thought - en route to Vaz Te, who cut the ball back for Nolan.
"I wasn't sure at the time but I saw a tape afterwards and it came off one of our lads so I can't fault the ref," Lambert said.
"We started well but it's all very well having lots of ball but not if you don't score.
"We need to be able to service Darren Bent. Losing Gaby (Agbonlahor) was a big blow to us before this game.
"You've got about three and a half milliseconds to hit the ground running in the Premiership."