Having secured promotion on Tuesday against Charlton, the Bluebirds were on their way to a first second-tier crown in their history when Craig Conway fired a first-half opener at Turf Moor.
David Edgar snatched a point for Burnley in second-half injury time but it would not stop Cardiff cracking open the champagne.
"It's absolutely fantastic, I'm delighted for everyone at the football club," said manager Mackay.
"It was obviously a tremendous feeling on Tuesday night but for us to go and win the title is something that we then spoke about within a day.
"We wanted to go and get the boys their medal and win the championship.
"We've been sitting there since late November and it's been an awful long time for us to be top and be chased and be examined. I think they deserve it.
"I'm delighted that we've had the mental strength and character to have that consistency and just keep going, being relentless. That's what we were this year - relentless.
"I spoke to them about (how) most people get to watch history happen; very few people get to actually make history.
"For them to do that today for the city of Cardiff is something that I'm very proud of."
Mackay stated earlier in the week that quality summer reinforcements would be needed if Cardiff are to make a successful transition to the Premier League.
But plenty of the players on display in the first half made sure their manager will have plenty to think about, none more so than South Korea international Kim Bo-kyung.
"I think everybody that's watched Kim this year can see that he's made to make an impression at that level," Mackay added.
"He's an international footballer and is one of the young stars of South Korea.
"As the season's gone on he's been more comfortable in his environment - he's learning English and he's been more adaptable.
"Anyone who can play in the Championship with that quality has got to have something about them and I think he's certainly one for the future."