They were first to open their account, when halfback Tian Meyer, celebrating his 50th appearance, capped a length-of-the-field breakout from their own goal-line. The visitors had looked threatening, but could not free the ball from a maul, just a few metres out from the try-line.
Number eight Henco Venter made a telling run from the back of the ensuing scrum, finding Sergeal Petersen down the flank and finally Meyer for the try.
First-five Fred Zeilinga could not convert, but extended the lead with a penalty soon after.
Counterpart Marty Banks finally had the Highlanders on the scoreboard with a penalty from a dominant scrum in Cheetahs territory.
After 23 minutes, Banks put the Highlanders ahead, fending off a weak tackle to cross under the posts for a try and easily converting for a 10-8 scoreline.
Squire had needed early medical attention for what seemed like a thumb injury and lasted just 10 more minutes, before leaving the game.
Moments later, Petersen had a try ruled out for obstruction, but the Highlanders lost fullback Matt Faddes to a yellow card, when he was deemed to have tackled late and with no arms. Replays suggested neither was true.
The Cheetahs immediately capitalised, sending Petersen over in the right corner for a try that Zeilinga converted from the sideline for a 15-10 advantage at the break.
Soon after halftime, the Highlanders were down to 13 men, after prop Siate Tokolahi was expelled for a dangerous tackle. They were lucky to keep Liam Coltman, when he collapsed an attacking maul, but could not withstand the rampant Cheetahs forwards, as they carried Torsten van Jaarsveld over for a try.
With Faddes back, but still short-handed, the Highlanders struck next, with tries to Rob Thompson and Waisake Naholo, both converted by Banks.
Ahead 24-22, the NZ side seemed to have weathered the storm, but were rocked by quick tries to Oupa Mohoje and Meyer, for his second, and suddenly found themselves 10 points in arrears.
Then the Cheetahs produced a spectacular try for prop Ox Ncha, who was somehow on the end of some skilful interpassing to stretch the margin seemingly beyond the Highlanders' reach.
Over the closing 18 minutes, they struggled to find the composure or cohesion to mount a compelling comeback, but got the help they needed, when Cheetahs replacement Uzair Cassiem was yellow-carded for a dangerous tackle.
Now on the front foot, the Highlanders quickly rattled on tries to Tevita Li and Faddes to close within a score, with seconds remaining.
They were not to be denied, driving within centimetres of the line on several occasions, before Naholo finally found the gap for the gamewinner.
"We were definitely put under the pump there are a few times," admitted skipper Luke Whitelock. "The guys believed over the last 10 minutes and really put it all out there, and we managed to get the result.
"We had nothing to lose, we just had to go out there and pull trigger. The bench guys came on and did some good stuff, and that was the difference really."
The result leaves the 2015 champions with seven wins and three losses for the season, further consolidating their grip on a wildcard spot with a weekend of results to come.
Highlanders 45 (Waisake Naholo 2, Marty Banks, Rob Thompson, Tevita Li & Matt Faddes tries; Banks six conversions & penalty) Cheetahs 41 (Tian Meyer 2, Sergeal Petersen, Torsten van Jaarsveld, Teboho Mohoje, Ox Nche tries, Fred Zeilinga four conversions & penalty)
