Conor Branson, Yusifu Ceesay and Bailey Hobson all netted in the closing stages to send Alfreton through, making it three wins out of four unbeaten games in all competitions.
And Heath said the late comeback made up for Alfreton getting out of the blocks slowly.
He said: “I’m obviously pleased with the outcome in getting through to the next round but I was disappointed with the first-half as we didn’t have our usual verve and competitiveness.
"We’ve then conceded a really poor goal before half-time but we made some changes which gave us a little bit more power and it was much better and suddenly we were in a game, although they missed a great chance to make it 2-0.
“The fight was there, we kept plugging away and the goals were good.”
Alfreton now head into a congested festive period where they will face two games in three days – first at home to Boston United on Boxing Day and then away at York City two days later.
They’ll then travel to Boston for the return fixture on New Year’s Day.
Heath said: “Christmas is always lovely because we always seem to be up against the big teams!
"Boston at home is the first one to look at and we’ll go into it with confidence.
"If we play the way I want us to play, with that verve, dynamism and grit amongst the lads then I believe we’ll give anybody in any league a big game.
"They’re tough fixtures because of what Boston and York are in this league or in any league.
"So it’ll be a really big week and it’ll help shape the future because if we can go into January and we’re in good shape in the table then we’ll be more than pleased.”
The draw for the fourth round of the FA Trophy has seen Alfreton handed a home tie against either Bradford Park Avenue or Heath’s former club FC Halifax Town, the two sides due to meet on Tuesday night after this week’s Derbyshire Times had gone to press.
The tie will take place on Saturday, January 15.