Saracens made a late change to their line-up as Olly Hartley, who had not been named in the squad, replaced Lucio Cinti.
And it was the hosts who made the brighter start, with Caluori instinctively picking off a loose pass from Slade and running through unchallenged, before Theo Dan set up Malins to touch down in the corner at the end of an effective line-out routine.
The Chiefs’ start was made worse when Tom Hooper was sent to the sin-bin for a tackle where he made contact with Dan’s knee. The fact the initial contact was with the thigh was what prevented referee Luke Pearce from producing a 20-minute red card.
Despite the numerical disadvantage, the Chiefs continued to fight and hit the front after a long TMO review.
Campbell Ridl won a 50:50 with Caluori after Ivan van Zyl let the Exeter box-kick go over his head.
The ball bounced up and hit Ridl in the face before he took control and set up Varney for an easy finish, with replays inconclusive as to whether the ball touched his hand for a knock-on in the process.
Ridl’s Prem debut was marked with a trip to the sin-bin for a deliberate knock-on that cut out Hartley’s pass and denied Caluori a second score, with the penalty try awarded for Sarries to retake the lead.
But Caluori eventually got the second try his performance deserved after Malins’ good play on the other side, with the ball worked across to Elliot Daly, who found his wing with a high, looping pass before Caluori dived into the corner with his last action of the match.
The best was left for last as two quick tries from Fisilau set up a grandstand finish.
First, he powered over at the back of a maul, before then weaving his way through to give the Chiefs a one-point lead.
Exeter survived a strong drive from the hosts as the defence managed to hold the ball up on the line, before Farrell missed a penalty attempt that would have retaken the lead for the Men in Black.
And with the final action of the game, Slade turned the ball over to sprint through into the corner and wrap up the points for the visitors.
