Exeter claimed top spot in the Premiership with victory against Northampton, who they will host next week in the play-off semi-finals.
Piers Francis and Henry Slade exchanged tries before Rory Hutchinson went over to give Saints the lead.
Exeter hit back with scores from Ben Moon and Don Armand, before Cobus Reinach made it 21-21 at half-time.
Tries from Matt Kvesic and Dave Dennis put Exeter in control and Luke Cowan-Dickie completed the win late on.
With Saracens losing 31-29 at Worcester, Exeter top the table in the regular season by eight points.
Northampton ensured a fourth-placed finish despite their defeat, after fifth-placed Harlequins were beaten 27-25 at Wasps.
Saints' one extra win was enough to see them through after the two sides ended the season tied on 56 points.
Exeter, who had stuttered in their last few outings, looked shaky at the beginning as Saints made the perfect start.
Jamie Gibson's run and deft dummy created the break that Francis finished off, but Chiefs quickly restored parity through Slade before Olly Woodburn had a great score chalked off by the television match official after a forward pass in the build up.
Reinach's excellent run gave Saints the field position and overlap they needed for Hutchinson to go over before Exeter took control and looked like the side, along with Saracens, that has dominated the Premiership this season.
Dave Ewers was held up on the line before Moon and Armand - making his 100th Premiership appearance - went over, and it appeared as though Exeter would get another try as intense pressure saw Courtney Lawes and Alex Waller sin-binned in the final minutes of the half.
But a loose pass from Ollie Devoto was gathered in by Reinach, who sprinted the length of the field to stun the home crowd.
Northampton battled bravely with reduced numbers before Kvesic went over for the bonus-point try after a catch and drive in the final seconds of Waller's yellow card, and Dennis got the fifth after a line-out.
Hutchinson had a second score ruled out after a replay showed he failed to ground the ball, while Chiefs replacement Cowan-Dickie was forced over for a third close-range effort of the day for Exeter.
But, after James Lang's last-minute long distance penalty for Harlequins just missed going over at Wasps, Saints went through by virtue of their 11 wins after ending the season level on 56 points with Quins and Bath - who both registered 10 victories in 22 matches.
Bath reached their total after winning 32-31 at Leicester - with 56 points being the lowest total for a fourth-placed side in Premiership history.
Exeter Chiefs director of rugby Rob Baxter:
"The first half was a bit of a stop-start affair, but once the game got going second half I thought we really came to the fore - there were so many boxes that we ticked really well.
"The flip side of that now is that double-headers are some of the hardest situations to deal with as a player and as a coach, and we've got to make sure that mentally we're bang on on Monday to make sure we're going to go out there and decide that we're going to fight for a final place.
"Northampton are a very dangerous side, they showed that numerous times today, so we're going to have to be prepared to work as hard again, if not harder, next week if we want to get to the final."
Northampton Saints director of rugby Chris Boyd:
"It's a bit like kissing your sister, it's bitter-sweet. We're disappointed to concede 40 points here, despite the fact they're a pretty decent side.
"We had to rely on Wasps doing a job for us, but in the end we're obviously very pleased to make the semi-finals for the first time since 2015.
"I'm pleased, but there'll not be much delight in the changing room, I can tell you that.
"If you compare our game here with the last time we played Exeter, we didn't create many chances in that game, but what we created we finished and we didn't allow them into the game because were were disciplined and accurate."
Exeter Chiefs: Nowell; Cuthbert, Slade, Devoto, Woodburn; J Simmonds, White; Moon, Yeandle (capt), Williams, Dennis, Hill, Ewers, Armand, Kvesic
Replacements: Cowan-Dickie, Keast, Francis, Skinner, S Simmonds, Maunder, Steenson, Hill
Northampton Saints: Tuala; Collins, Hutchinson, Francis, Naiyaravoro; Biggar, Reinach; Waller (capt), Marshall, Hill, Moon, Lawes, Gibson, Ludlam, Harrison
Replacements: Fish, van Wyk, Franks, Ribbans, Wood, Mitchell, Grayson, Burrell
Sin-Bin: Lawes (40 mins), Waller (40 mins)
Referee: Karl Dickson.