Hibernian missed a first half penalty, before Nicholson struck from range for Hearts and Prince Bauben doubled the lead from the penalty spot.
"I don't think we dominated totally, I thought it was a typical derby," Neilson told Sky Sports.
"There was some football played, which was nice to see, but it's just great to get the win. Sam has got unbelievable ability, we know if we can get him into those areas he can produce that kind of stuff."
Hearts' next fixture is away to Livingston in the Challenge Cup on Wednesday.