Ronald Koeman's men were handed a golden opportunity to take the lead on the hour at Turf Moor when George Boyd was penalised for a trip on Ryan Bertrand.
Dusan Tadic stepped up, but saw his spot-kick palmed away by Heaton, before Ashley Barnes then sealed the win for the hosts 17 minutes from time.
And Dyche hailed the performance of his keeper, before claiming that the level of quality within his side was there for all to see during the victory, which lifts Burnley out of the Premier League drop zone.
"There was some real quality first half and second half they got the upper hand and a penalty which sees a fantastic and massive save from Tom," he said.
"After that, we responded really well and got back on the front foot and got back into the game.
"The players have given everything again today and I say again because I think they do every week for myself and the fans of the club.
"I've said all along that that's what I believe in and there is some quality within that of course.
"But that will, that desire, that energy and their authenticity today to make sure we got a result was on show as much as it has ever been."
Dyche went on to say he expects Burnley to have to scrap for their Premier League lives for the rest of the season, despite a run that has seen them lose just one of their last six top-flight fixtures.
He added: "We're going to have to fight for every inch in every inch to make sure we get what we want this season.
"We're showing great signs of that at the moment. Five clean sheets in 15 matches is another powerful marker."
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