Harlequins' 27-25 defeat at Wasps, in which Quins fly-half James Lang missed a late long-distance penalty, meant Saints stayed fourth on an eventful final day of the regular season.
And that means Chris Boyd's men will again meet table-topping Exeter in a shootout for a place at Twickenham next weekend.
Saints went ahead twice against Exeter during the first half thanks to some slick play that culminated in tries for Rory Hutchinson and Piers Francis.
Exeter eventually gained some control, going 21-14 up and forcing Courtney Lawes and Alex Waller into the sin bin.
But Saints scored a remarkable try to level before half-time as Cobus Reinach intercepted inside his own 22 and sprinted in to score with his side still down to 13 men.
However, Saints could not score a point in the second half as Exeter dotted down twice more to win it.
It meant the black, green and gold finished fourth thanks to earning one more win than Quins during the regular season.
And Saints will now bid to turn the tables on Exeter in the semi-final next week, while Saracens will host Gloucester in the other final-four game.
Saints had started superbly at Sandy Park and Jamie Gibson unlocked the Exeter door as he looked like he would send the ball right before switching and going left, allowing Ahsee Tuala to gallop into space before Francis picked up at the breakdown and scored.
Dan Biggar converted and the noisy home crowd had been silenced inside the opening four minutes of the match.
Saints then lost hooker Reece Marshall to a left leg injury, bringing James Fish into the action after just five minutes.
Exeter had taken a few minutes to spring into life, but they were back level after Henry Slade glided in behind the posts and Joe Simmonds converted.
The Chiefs thought they'd scored again soon after as Ollie Woodburn sped around Tom Collins and dived over under the posts, but the pass to Woodburn was rightly deemed forward and the try was ruled out.
Saints were having plenty of joy out wide, with Taqele Naiyaravoro causing problems on the left in what was an open game during the formative stages.
Tuala had looked lively and after he escaped Exeter clutches close to halfway, Cobus Reinach carried forward at speed and Hutchinson eventually finished with a trademark flourish, even running in behind the posts to make the conversion easier.
Biggar did the rest and Saints led 14-7 having more than given what they had got from their hosts.
Exeter looked likely to respond again and after were held up at the last by Waller, Ben Moon finally got the job done, making the most of his side's power and persistence.
Simmonds converted as Saints lost Biggar to a blood injury, meaning James Grayson came on and immediately booted the restart straight out of play.
Exeter won a penalty at the resulting scrum, Slade kicked it to the corner and Exeter worked the ball well, with flanker Don Armand finishing things off.
Simmonds converted to make it 21-14 at the break and suddenly Saints were all at sea as Alex Cuthbert stormed forward and Lawes was sin-binned after feeling he had legally turned the ball over.
Exeter took the scrum in front of the posts and they put Saints under huge pressure, forcing Waller to infringe, leaving his side two men down after he was yellow carded.
It looked like the Chiefs were certain to score but they lost their concentration and Reinach pounced, intercepting a poor Ollie Devoto pass and sprinting almost the length of the pitch to score.
Grayson converted and the scores were level after a frantic first half.
Exeter still couldn't find a way through before Lawes returned from the bin as the home side's sloppy play continued to thwart them, while Saints were stepping up defensively.
But eventually Exeter made their man advantage pay as Matt Kvesic broke from a lineout and dived over in dramatic fashion.
Simmonds missed the conversion but the Chiefs were now on the charge and Saints had to do some strong defensive work on their own line to stop the hosts scoring again.
But it wasn't long before Exeter got the try they wanted as they caught Saints cold with lock Dave Dennis spotting a gap and sprinting over the line with one hand in the air in celebration.
Simmonds converted and the Chiefs held a 33-21 lead with more than 20 minutes to go.
But Saints were not done and Hutchinson almost scored a lovely try, kicking ahead before just being unable to get the ball down over the line.
The action then petered out slightly but there was still time for another Exeter score as Luke Cowan-Dickie rumbled over, with Simmonds again converting to make it 40-21.
And Exeter then saw it out with ease as they secured top spot in the table, setting up a reunion with Saints next Saturday.
Exeter Chiefs: Nowell; Cuthbert (S Hill 53), Slade (Steenson 74), Devoto, Woodburn; J Simmonds, White (Maunder 74); Moon (Keast 65), Yeandle (c) (Cowan-Dickie 55), Williams (Francis 50); Dennis (Skinner 59), J Hill; Ewers (S Simmonds 60), Armand, Kvesic.
Saints: Tuala; Collins, Hutchinson, Francis (Burrell 53), Naiyaravoro; Biggar (Grayson 59), Reinach (Mitchell 46); Waller (c) (van Wyk 61), Marshall(Fish 5), Hill (Franks 61); Moon (Ribbans 57), Lawes; Gibson, Ludlam (Wood 57), Harrison.
Referee: Karl Dickson