Mullins moved past the great Dermot Weld with career win number 4,378, even though the latest star to emerge from his Closutton production line was a little below par.
A couple of sloppy jumps left Paul Townend nudging the 1-5 market leader on the turn for home, but he soon reeled in front-running stablemate Mirazur West once straightening up.
Jessica Harrington’s Jetara came out of the chasing pack to move into second place, but Ballyburn produced a fine leap when it mattered most at the last to quickly ease any fears of an upset.
Although not as impressive as when powering home in the Gallagher Novices’ Hurdle at Cheltenham, the six-year-old completed a hat-trick of Grade One triumphs with three and a quarter lengths to spare.
On his record-breaking feat, Mullins said: “Dermot pioneered going down to Australia and won the Belmont Stakes, an American Triple Crown race. He’s done superb things, so to be mentioned alongside him is fantastic.”
Mullins sent out his first winner in 1988 and has become the dominant force in the Irish training ranks, with the Closutton handler set to be crowned champion for the 18th time when the Irish season concludes on Saturday.
His latest landmark comes at the end of a truly memorable season that saw Mullins saddle his 100th winner at the Cheltenham Festival and become the first Irish trainer since the great Vincent O’Brien some 70 years ago to win the British trainers’ championship.
His big-race haul included the Cheltenham Gold Cup with Galopin Des Champs, the Champion Hurdle with State Man and the Grand National with I Am Maximus.
He added: “It’s tremendous to break Dermot’s record. Dermot broke lots of records and is a super trainer. Any time you can do something that Dermot Weld did, I’m in very privileged company.
“I imagine someone else will come and break those records, with new fixtures every year. Numbers and records are there to be broken and it’s just great to be here in this position.”
As for Ballyburn’s performance, Mullins commented: “He did it well, he’s just got a huge engine.
“I’m looking forward to whatever he’s going to do next year, whether we’ll stay hurdling or go jumping fences. I’ll have to have a word with connections, David Manasseh and Ronnie Bartlett.
“He’s just a really exciting prospect, the way he got down there and raced over the last two furlongs.
“It looked for a stride or two ‘would he get beaten’, as Jetara was coming with a lovely run. We were delighted with the way our fella finished.
“He’s still not mature and he’s racing too much throughout the race. Once he learns to settle a bit more and use his power at the end of the race, I think he’s going to be a super horse.
“We can go either direction with him. Two miles, two and a half – and on his pedigree, he can go out to three miles.
“I was just chatting to his point-to-point rider and he said he’s a fabulous jumper of a fence, so we have all that to put into the mix and see what direction we go in the future.
“A few of ours ran a bit flat earlier in the day and we were wondering would this fella do the same, but he didn’t. Good horses just dig deep and they do it.”