Højgaard held off Nacho Elvira after coming from seven shots behind to finish level with the Spaniard on 13 under par.
“I’m speechless at the moment," Højgaard said afterwards. "To be fair, I didn’t think I’d end up in a play-off – it’s quite amazing. To win a home event, it’s amazing.
“It isn’t that long ago that I was one of those kids trying to get balls and signed gloves, so it’s cool to see them out here supporting us.
“This is better than I could ever dream of. We’ve wanted a Danish winner for so long now, so to be the first one to do it is amazing.”
It was Højgaard's fourth DP World Tour win and extended his perfect record in play-offs, having won the 2019 AFRASIA BANK Mauritius Open and the following year’s ISPS HANDA UK Championship in the same manner.
Kalle Samooja and Højgaard had both moved within a shot of the lead alongside Ramsay, Højgaard with four birdies in six holes on the front nine while Samooja birdied three of the first five.
MacIntyre birdied the fourth to take the lead as Elvira’s birdie putt lipped out as he was already shaping for a celebratory fist-pump.
Both players birdied the par-five fifth but Højgaard and Ramsay came back at them.
Elvira saw another putt horseshoe out, a tremendous left-to-right effort from the fringe of the ninth green, where MacIntyre hit his approach to two feet to secure a birdie and a two-shot lead at the turn.
Elvira hit back at the tenth but when he bogeyed the 13th and MacIntyre ran up a triple after finding trouble down the right, Højgaard’s long birdie putt at the last saw him sharing the lead with Ramsay and Elvira.
Ramsay birdied the short 16th to go one shot ahead and MacIntyre and Elvira both bogeyed the 15th – though the Spaniard responded by birdieing the 16th and sneaking an 11-foot birdie putt at the next into the side of the hole.
Ramsay found the left-hand rough off the final tee and then put his second in the water hazard, the resulting double-bogey meaning a third-place finish at 12 under.
That opened the door to Elvira, whose tee-shot split the fairway, but he went through the green and mishit his third from the fringe of a bunker, needing a long two-putt just to earn a play-off.
Højgaard got up and down from the bunker to par the first play-off hole and take it to a second, where Elvira did likewise.
Højgaard’s pin-high approach third time round gave him the advantage but his putt shaved the hole, and they shared the next two holes as well.
At the sixth extra hole, though, Elvira flew his second shot out of bounds into the practice grounds leaving Højgaard to two-putt for a victory which lifts him from 67th to 12th on the Race to Dubai Presented by Rolex.
Elvira had some consolation as he took one of the took the three Open Championship places on offer to the highest finishers not already exempt. The others went to Samooja and Marc Warren, who shared fourth place with MacIntyre and Alexander Björk on 11 under.
Matthew Southgate shot 61, the best round of the week, to finish nine under. The Englishman was threatening a 59 after a run of five straight birdies to the 17th but, needing another at the last, he instead sent his tee-shot into the water and made a bogey five.