The Englishman started the day seven shots off the lead but after turning in 33 on the Earth Course, picked up a shot on every hole from the tenth onwards to sign for a stunning career-best 60.
He was two feet from holing a bunker shot for an eagle and a 59 on the last but had to settle for the lowest round of the season, a 16 under total and a one-shot lead as he looks for a first Rolex Series title.
Playing partners Tommy Fleetwood and Viktor Hovland were the closest challengers after continuing to match each other with rounds of 69-66-66 in Dubai.
Dane Jeff Winther was then at 14 under after a 64, a shot clear of Scot Ewen Ferguson who carded the same score and another Dane in overnight leader Nicolai Højgaard, who signed for 70.
Wallace's nine birdies in a row, 12 birdies in a round, nine-hole score of 27 and nine-hole score to par of nine under all match DP World Tour bests but will not go in the record books due to preferred lies being in place.
The 32-year-old only made it into the season-ending event courtesy of his finish at last week's Nedbank Golf Challenge but now has a chance to claim a fifth DP World Tour title and first since the 2018 Made In Denmark, which was his fourth in 16 months.
He has not missed a cut in a regular DP World Tour event in 2023 but was in danger of losing his card before he finished second at the D+D REAL Czech Masters, a scenario that seems barely believable as he sits on the cusp of the biggest win of his career.
"What a day, an amazing day," he said. "I just tried my hardest to get myself back into the tournament. I'm really happy that I've been able to do that. Played great.
"At the end, I didn't even think there was a 59. Honestly I think it helped me a little bit. I just played nicely coming down the stretch, just keep getting one more if I could and I managed to do that.
"In the Czech Republic, I was playing to keep my card, played mostly on the PGA TOUR this year," added the 2023 Corales Puntacana Championship winner.
"So once I came second there, it kind of boosted me and then I wanted to make DP World and now I'm here and we have a chance to win tomorrow. That's the full focus there."
Wallace turned in 33 thanks to a two-putt birdie on the par-five second, a tee-shot to ten feet on the sixth and a 13-footer on the eighth.
A 29-foot putt on the tenth then kickstarted his remarkable run as he holed another long putt on the next and left himself six feet on the 12th.
A stunning tee-shot to three feet on the 13th was followed by a second two-putt birdie of the day and when he pitched to two feet at the short 15th, he was just one back.
An approach to tap-in range at the 16th had him in a share of the lead and a tee-shot to six feet on the 17th meant the magic number was in sight with an eagle on the last.
He then hit a perfect drive on the par-five 18th but found sand with his second before leaving his bunker shot just two feet from the pin as he made a closing birdie and matched the lowest round in Rolex Series history.
Fleetwood joined the early lead with a seven-footer on the first and birdie on the second after leaving himself nine feet for an eagle and led by two following a hat-trick of gains from the fifth courtesy of two smart iron shots and another two-putt birdie.
The Englishman dropped a shot on the 12th but was back in a share thanks to a 58-foot eagle putt on the 14th and a birdie on the 16th but made a bogey with a three-putt on the next.
Norway's Hovland birdied the first and second in similar fashion to his Ryder Cup team-mate before getting up and down on the seventh, leaving himself four feet on the tenth and 13th and playing a beautiful approach into the 16th in a bogey-free effort.
Winther holed out from 143 yards for an eagle on the 11th and made another on the last to go with six birdies and two bogeys, while Ferguson's 64 contained an eagle and six birdies.
Højgaard bogeyed his last two holes to fall back and sit one shot ahead of Swede Jens Dantorp and Frenchman Matthieu Pavon and two clear of twin brother Rasmus, defending champion Jon Rahm, another Dane in Thorbjørn Olesen, South African Thriston Lawrence and France's Antoine Rozner.