Northampton were denied an Investec Champions Cup semi-final spot for the third successive season after Bath won an all-time tournament classic 43-41 following a late yellow card for Saints' England star Henry Pollock.
Bath triumphed following a mesmeric 11-try encounter at the Recreation Ground, and they will now face holders Bordeaux-Begles or six-time champions Toulouse.
The winning score came via a Ted Hill touchdown that Finn Russell converted three minutes from time - and shortly after Northampton went down to 14 players when Pollock departed.
Attacking rugby was showcased in all its glory as the top two teams in this season's Gallagher Prem cut loose, with Northampton posting five first-half touchdowns from Pollock, Fraser Dingwall, Tom Lockett, Josh Kemeny and Ollie Sleightholme, all converted by Fin Smith.
It looked as though two Smith penalties after the break would thwart a Bath fightback as drama continued right to the end, but it was not to be following Bath tries for Hill, Kepu Tuipulotu, Finn Russell, Henry Arundell, Francois van Wyk and Tom Dunn, while Russell booted five conversions and a penalty.
Northampton flew out of the blocks, capitalising on a third-minute yellow card for Bath number eight Miles Reid by scoring immediately when Pollock smashed his way through three defenders for a try that Smith converted.
Bath were all over the place, with Saints cutting them open just four minutes later as scrum-half Archie McParland's one-handed pass created space for Smith and Tommy Freeman before Dingwall finished off a scintillating move.
Smith again added the extras, but Bath's response proved a rapid one when Dunn took a quick penalty and crossed from close range. Russell's conversion made it 14-7 halfway through a breathless opening quarter.
And there was no end in sight to the try spree, particularly from Saints' viewpoint, as Lockett collected Smith's long pass for their third touchdown, then Kemeny completed a sweeping 60-metre attack that had its origins from a charged down Bath kick inside Northampton territory.
Smith booted two more conversions, and after Russell's opportunist try that he also converted, Saints struck again through Sleightholme, who cashed in on poor Bath defending to take his team merrily past 30 points.
With Smith maintaining his 100 per cent conversion rate, Bath were leaking scores at an alarming rate, yet they regained their composure and took the first-half try-count to a staggering nine when Arundell and Van Wyk crossed in quick succession, making it 35-26 at the interval.
Bath's England flanker Sam Underhill failed a head injury assessment just before the break, and Northampton quickly got the scoreboard moving again through a Smith penalty.
But even though they held a 12-point lead with 25 minutes left, Saints knew there was still much work to be done, and Freeman made a try-saving tackle on Bath captain Ben Spencer before Northampton lock JJ van der Mescht was sin-binned following repeated infringements.
And he had barely left the pitch when Tuipulotu powered over for a well-worked touchdown, setting up a fittingly thrilling final quarter to an epic encounter.
Smith and Russell exchanged penalties as the clock ticked down, then Pollock saw yellow and Hill's late show unforgettably sealed the deal.
