Northampton were denied a Champions Cup semi-final spot for the third successive season after Bath won 43-41 following a late yellow card for England star Henry Pollock.
Bath turned to a quartet of their England backs to turn the tide as they edged into the Champions Cup quarter-finals with a 31-22 victory over Saracens.
Bath finished top of Pool 2 in the Champions Cup with a ruthless 63-10 victory over Edinburgh.
Edinburgh staged a second-half comeback to pick up a hard-fought 26-24 win over Gloucester in the Champions Cup and consigned the visitors to a fifth straight defeat.
Bath lost a 26-point lead in an epic south-west derby only to snatch back victory in the most dramatic fashion in the closing seconds to beat Exeter 33-26, before Northampton Saints returned to the top of the Prem table.
Northampton stunned the Prem Rugby champions Bath with a devastating display of running rugby featuring a hat-trick of tries by freewheeling England back Tommy Freeman.
An astonishing start which saw Bath 28-0 ahead inside 19 minutes proved crucial as the Prem side saw off Munster 40-14 in their Champions Cup opener.
Bath snatched a 36-29 victory over Saracens that propelled them two points clear at top of the Gallagher Prem.