The Chiefs full-back caught fire in the first half at Melbourne's Rectangular stadium, securing a bonus point for the visitors in the 27th minute with his third try, and finishing a 24-point game with three conversions and a penalty.
Inspired by Wallabies backs James O'Connor and Kurtley Beale, the Rebels pulled within a converted try with five minutes left, but the Chiefs held on to retain top spot in the New Zealand conference and be well-placed for a run to the play-offs.
"Pretty relieved," Chiefs captain Craig Clarke said in a pitchside interview.
"Fair play to the opposition for playing the style they do."
Anscombe was denied a magical fourth try in the 78th minute by the television match official, who ruled the ball had been pushed forward by a team-mate just before the full-back received it wide and bolted for the left corner.
The Chiefs took a 24-14 lead at half-time, with Anscombe scoring his first try in the opening minute and inside centre Bundee Aki making it 10-0 in the 10th.
Scott Higginbotham put the Rebels on the board with a try five minutes later, but Anscombe responded with two tries in seven minutes, the second courtesy of a brilliant break from outside centre Tim Nanai-Williams.
The Rebels were kept in the game by towering lock Hugh Pyle, who scored the first of a brace of tries shortly before half-time, but the Chiefs sprinted away to a 31-14 lead 12 minutes after the break with a try by flanker Sam Cane.
Beale came off the bench eight minutes into the second half, making his return after a six-week exile from the Rebels.
He was stood down in March for punching two of his team-mates in an alcohol-fuelled incident on the team bus in South Africa, and appeared a bundle of nerves as his first kick went out on the full on the restart following Cane's try.
He was soon brought into the match by fellow Wallabies back O'Connor, and helped set up scrum-half Nick Phipps for a 68th minute try in the left corner with a brilliant break at midfield.
Big lock Brodie Retallick pushed the Chiefs' lead out to 39-26 by scoring an enterprising try with 10 minutes to play after charging down a clearing kick by James O'Connor.
But that only set up a grandstand finish as newly slimmed-down Beale tore through the Chiefs defence, running wide then cutting back inside to score a brilliant individual try in the 75th minute.
Rebels winger Cooper Vuna raised the roof at the 'Stockade' with a dash along the left touch-line in the 79th minute but was dragged into touch just short of the line.
The Rebels stole back possession with a poor line-out throw but were unable to penetrate the Chiefs' last line of defence in a frenetic finish after the siren.