The reigning Premiership champions were under all sorts of pressure during the first period, despite Saints losing David Ribbans to a red card after just 13 minutes.
Chris Boyd's side were up and at 'em during a tumultuous first 40 minutes, which ended with a standing ovation from the 4,000 fans in attendance as Exeter found themselves 18-0 down.
Saints' intensity levels had been through the roof as they continued to pour forward despite having a man fewer, and Rory Hutchinson put the finishing touches to two tries.
But Exeter came out firing after half-time as Saints fell slightly flat, with the Chiefs flying back into the game.
Saints refused to be overawed as Mike Haywood put them 26-19 up, but there were few ways out as Exeter ploughed on, earning a 29-26 win thanks to a Joe Simmonds penalty seven minutes from time.
Saints had been forced to make a change to their bench before the game as Paul Hill was ruled out with a neck problem, bringing Oisin Heffernan onto the bench.
Exeter also had to make an alteration with Scotland lock Jonny Gray failing a late fitness test, meaning Sean Lonsdale started in the second row and Don Armand was added to the list of replacements.
It was Saints who started the stronger, using an early scrum penalty close to halfway to put real pressure on deep in the Exeter half.
After a patient and precise move, Dan Biggar kept his cool with a tidy final pass for Hutchinson to score.
Biggar missed the conversion from out wide and Exeter were soon at the Saints door, but the home defence held firm to force a knock on and then a free-kick.
However, there was to be a huge flashpoint soon after with the 13th minute of the match certainly proving unlucky for Saints as Ribbans was sent off for shoulder contact to the head of Luke Cowan-Dickie.
Exeter would have been expected to grab the game by the scruff of the neck at that point, but it was Saints who lifted their levels.
Courtney Lawes won a huge penalty inside his own 22 and then Saints turned on the class at the other end, with Biggar and Hutchinson combining superbly for the Scotland centre to score.
Biggar converted to make it 12-0 and the game was being played at an incredible intensity, similar to play-off semi-finals and finals of years gone by.
Saints came agonisingly close to a third score as Matt Proctor, who has started with the bit between his teeth, flew towards line but the move ended when Taqele Naiyaravoro's attempted offload went astray at the last.
Saints still kept coming and it took an Exeter steal on their own line to stop the black, green and gold from putting together another scoring passage of play.
Lawes was running around like Fagin, stealing everything from Exeter as they frantically tried to get a foothold in the game.
But it was Saints who had all the pressure in the opposition 22 and Exeter were fortunate to escape a yellow card of their own after Tom James was on the end of a high tackle.
Biggar did make the Chiefs pay with a successful penalty though, stretching the lead to 15 points.
It was soon 18 as Biggar slotted a drop goal to reward yet more Saints pressure in Exeter territory.
The Chiefs had one final chance in a half of few for the champions, but Saints made up for losing a lineout with the clock in the red as they held out yet again to ensure their opponents would not score a first-half point.
Exeter soon made their mark after the restart though, staying patient and putting Sam Skinner in for the score.
Joe Simmonds hit the post with the conversion to leave the gap at 13 points but Saints were aware that the Exeter revival was underway.
It was soon two scores for the Chiefs as they pieced together a tidy move that was eventually finished by flying full-back Stuart Hogg.
Simmonds made no mistake with the conversion this time, and suddenly the deficit was down to just six points.
Saints then managed to deliver a huge scrum on an Exeter put-in, earning a penalty, which Biggar kicked to provide some welcome breathing space.
Exeter just kept coming though and replacement scrum-half Stuart Townsend made an instant impression, scoring after his side set up a lineout in Saints territory.
Simmonds converted and now the gap was just two points, but Saints came roaring back, with their replacements playing key roles.
Manny Iyogun and Alex Mitchell both delivered important passes before Freeman's fine work allowed Haywood to charge in out wide.
But Exeter responded almost immediately as Townsend danced his way through and offloaded to the onrushing Ollie Devoto for the score.
Simmonds converted to level the scores at 26-26 and the Saints fans tried to summon another rally from their players.
But the Saints penalty count was starting to go up and Exeter took the lead for the first time with seven minutes to go as Simmonds slotted a kick from the tee.
Saints kept fighting right to the final whistle, applying pressure on Exeter inside their own half before a forward pass was called and a sensational game of rugby ended.
Saints: Freeman; Sleightholme (Dingwall 24), Proctor, Hutchinson, Naiyaravoro; Biggar (Grayson 61), James (Mitchell 57); Waller (cc) (Iyogun 59), Matavesi (Haywood 59), Painter (Heffernan 65); RIbbans, Ratuniyarawa (Moon 57); Lawes (Coles 68), Ludlam (cc), Wood.
Exeter Chiefs: Hogg; Cordero (Devoto 51), Slade, Whitten, O'Flaherty; J Simmonds (c), Maunder (Townsend 57); Moon (Hepburn 50), Cowan-Dickie (Yeandle 57), Williams (Street 62); Lonsdale, Hill; S Skinner, Kirsten (Armand 62), S Simmonds (Capstick 64).
Referee: Craig Maxwell-Keys