The duo may be 27 years apart in age but there was nothing to separate them at Al Mouj Golf and they both reached nine under after 36 holes, one shot ahead of Belgian Nicolas Colsaerts and Finn Kalle Samooja.
Danish 18-year-old Højgaard became the third youngest winner in European Tour history when he won the AfrAsia Bank Mauritius Open earlier this season and he carded a 68 with an eagle, five birdies and three bogeys.
Gallacher - whose son and caddie Jack was born nine days before Højgaard - has four European Tour wins in his 45 years and he went one better with a bogey free 67 - his first blemish free round since the 2018 Alfred Dunhill Links Championship.
Gallacher's victory at last season's Hero Indian Open was his only top ten on the 2019 Race to Dubai but the Scot feels his game is trending in the right direction.
"I’m delighted because it was tough from the get go this morning," he said. "This is a great golf course and one of the best we play all year but it’s very demanding and you’ve got to play good golf. If you don’t you’ll get found out and fortunately I’m playing pretty well at the moment.
"I had been struggling a bit recently so I changed some things up and I’ve worked hard on my game. Things started to turn around at the Saudi International where I started to hit some good shots and it’s amazing how it can snowball after that.
"You start to chip and putt a bit better and start to compete and hopefully by Sunday you just want to be in with a chance coming down the stretch to win. It’s a good spot to be in but there’s still a lot of golf to be played over the weekend."
Højgaard is playing just his tenth European Tour event and was delighted with his putting display as he held a 36 hole lead for the first time.
"You can’t avoid bogeys out here so I knew at some point I would drop a shot because in this wind it’s tricky to keep the ball on the greens and fairways," he said. "But I managed to bounce back with a few birdies.
"I’m putting well at the moment and I holed some long putts today which always helps - it’s always a bit of a bonus when you make a few of them and that’s been the key so far."
Højgaard made a flying start from the tenth, eagling the par five 12th to leapfrog overnight leader Migliozzi before making birdies on the 14th and 15th to get to nine under.
Bogeys on the 17th and 18th stalled his progress and he was in a share of the lead with Gallacher, who birdied the fifth seventh and eighth to turn in 33.
Gallacher's playing partner Samooja also turned in 33 as he recovered from an opening bogey with gains on the second, third, sixth and eighth.
I’m putting well at the moment and I holed some long putts today which always helps - it’s always a bit of a bonus when you make a few of them and that’s been the key so far - Rasmus Højgaard
Højgaard holed a 20 footer on the first to get to eight under but he soon had company as Samooja birdied the tenth and 11th and then holed a 15 foot putt for eagle on the next.
Gallacher got up and down from the sand on the 12th to make it a three way tie and a long putt on the next put him ahead alone before Samooja followed him in.
Højgaard three putted the third but picked the shot back up from 18 feet on the fifth, before a two putt birdie on the par five seventh had him back in a share.
Samooja found water on the 15th to surrender a triple bogey but got a shot back on the par five next and holed a long double breaker on the last to get back within one.
In the afternoon, Amundi Open de France champion Colsaerts started with a bogey but then made a hat-trick of gains from the third before giving back a shot on the seventh to turn in 35.
He had a good eagle chance on the 12th but settled for a birdie before holing a very long putt on the 13th, birdieing the next and taking advantage of the par five 16th to get within one.
Luiten had reached eight under but dropped two shots in his last three holes in a second consecutive 69, while Rock was bogey free in an excellent 65.
McEvoy carded a 68, Stone signed for a 71 and Migliozzi registered a 72.
Germany's Martin Kaymer recorded the first hole-in-one in tournament history and the fifth of the 2020 Race to Dubai when he holed a pitching wedge from 162 yards at the 13th.