George Pisi scored a dramatic late try to shoot down Exeter and send Saints to the top of the Aviva Premiership table.
The Samoan centre popped up with three minutes to go to break the Chiefs resistance and Stephen Myler kept his nerve to kick the conversion and earn a 17-16 success.
Exeter had looked set to claim a gritty victory in the swirling wind at Sandy Park after Dave Lewis dotted down in the 53rd minute.
Gareth Steenson took his personal tally to 11 points with the conversion, giving Exeter a 16-10 lead that they looked set to hold on to.
But the Saints pack turned on the power in a series of late scrums, giving Pisi the platform to find a gap and a crucial score.
James Wilson and Samu Manoa registered in the first half of a win that sees Saints take over at the summit from Saracens, who slipped up against London Irish earlier in the day.
Saints knew they would have to work hard for a win at Sandy Park, with eight internationals missing.
And they started brightly, with line breaks from Kahn Fotuali’i and Phil Dowson putting the home side under immediate pressure.
Exeter were a man down inside seven minutes as skipper Dean Mumm was punished for stamping on Sam Dickinson. It could easily have been a red.
But the home side belied their numerical disadvantage as Alex Waller was penalised for a high tackle and Steenson bisected the posts with the penalty.
He failed with a more difficult attempt soon after and Saints were able to register before Mumm returned from the bin.
Some patient build-up play saw Manoa send Wilson over for a tidy try, with Myler missing a tricky conversion attempt that got caught in the strong breeze.
Saints were threatening and they were agonisingly close to a second score as Dickinson’s powerful surge through the middle was halted at the last.
Exeter responded, but saw another Steenson penalty drift wide of the posts.
Saints were hit with a blow as Christian Day was forced off with an injury, with GJ van Velze on in his place, and Exeter were soon in front as Steenson found the target.
Manoa wasn’t going to let things slip that easily, though, and he pounced in the corner for his ninth try of the season.
The big American was saluted with his now trademark chant from the small group of travelling Saints fans and reacted with a smile and a raised hand.
Myler again fell short with the conversion in outrageous kicking conditions, before Steenson slotted a much easier effort at the other end to bring Exeter back to within a point.
And they were soon a man up after the break as van Velze was sent to the bin for obstruction.
Saints were threatening to shoot themselves in the foot as Wilson didn’t look when offloading on halfway and Exeter raced clear, with Dave Ewers eventually held up.
But the Chiefs didn’t have to wait long for their try as Lewis dived over and Steenson converted to put his side 16-10 up.
Saints were struggling to make ground against the blue tide, which was relentless in its refusal to allow the opposition to enter the home 22.
But some good game management from Myler helped earn some territory and after a series of big scrums from the away pack, Pisi showed his intelligence to score.
Myler kept his nerve to register the conversion and secure the most dogged of successes for his side.
Exeter: Arscott; Jess, Whitten, Dollman, James; Steenson, Lewis (Thomas 65); Moon, Yeandle (Cowan-Dickie 61), Brown (Tui 54), Mumm (c) (Armand 66), Welch, Ewers, Scaysbrook (Johnson 53), Horstmann.
Replacements: Rimmer, Slade, Vainikolo
Saints: Wilson; K Pisi, G Pisi, Waldouck (Hooley 67), Elliott; Myler, Fotuali’i; A Waller (E Waller 68), Haywood, Ma’afu (Denman 52), Manoa, C Day (van Velze 32), Clark, Dowson (c), Dickinson.
Replacements: McMillan, Nutley, A Day, Autagavaia.
Referee: Greg Garner