Tigers' 30-24 triumph means they could confirm a play-off place if they beat Welford Road visitors Worcester on April 30 after tries from hooker Harry Thacker, wing Vereniki Goneva, and Freddie Burns underpinned victory.
Fly-half Burns added three penalties and three conversions for a 20-point haul, with Tigers' victory also nudging Exeter closer to the play-offs alongside confirmed qualifiers Saracens and Wasps.
If Leicester see off the Warriors later this month - or win their final regular season fixture at Bath seven days later - then Saints will be resigned to a fifth-placed finish at best, whatever they do in their remaining matches.
They gave as good as they got for most of an enthralling Franklin's Gardens encounter, with number eight Teimana Harrison and centre George Pisi scoring tries, while fly-half Stephen Myler booted three penalties and a conversion and JJ Hanrahan added a late penalty, but European Champions Cup semi-finalists Leicester possessed a composed edge at critical moments.
Saints remain firmly on course for a top-six finish and Champions Cup qualification next term, but their probable play-off failure will mean a season of under-achievement.
Leicester were forced into a back-row reshuffle after illness sidelined Brendon O'Connor, with Lachlan McCaffrey moving to openside flanker duties and Opeti Fonua called up at number eight.
Northampton showed four changes from the side that suffered a European Champions Cup quarter-final defeat against Saracens last weekend, including returns for England forwards Tom Wood and Kieran Brookes, but Red Rose skipper Dylan Hartley (concussion) and Wales wing George North (knee injury) remained sidelined.
Early exchanges were littered with errors, before Burns kicked a long-range penalty, and Northampton were emphatically undone two minutes later by a moment of Thacker brilliance.
Thacker beat two players just inside Saints' half, then cut back inside on an angled run to the posts before throwing a dummy pass that bemused Saints full-back Ahsee Tuala and he crossed between the posts.
It was a spectacular effort, with Burns' conversion opening up a 10-0 lead to leave Saints with plenty to ponder as Tuala went off injured and was replaced by Harry Mallinder, son of Saints rugby director Jim.
But Northampton responded impressively, drawing level through a 10-point burst in just four minutes that was started by a Myler penalty before Harrison rounded off a high-class attack when he capitalised on centre Luther Burrell's decisive midfield surge.
Myler converted, yet Burrell became the Northampton villain just three minutes later as a breathless first-half continued at pace.
Saints were attacking just inside Leicester territory through skipper Wood, only for the supporting Burrell to throw out a speculative pass that landed in Goneva's hands, and he sprinted 45 metres on an unopposed run to the line, with Burns' conversion making it 17-10.
Northampton continued to shade the second quarter, though, and a Myler penalty on the stroke of half-time reduced the deficit and set up an enthralling second period.
A second Burns penalty cancelled out Myler's strike, but Northampton moved to within two points of their opponents when sustained pressure reaped its reward as Jamie Elliott's pass found support near Leicester's line, and George Pisi finished off.
The game continued to ebb and flow in pulsating fashion, and after Burns completed his penalty hat-trick, the number 10 turned try-scorer when he finished off a flowing move that owed everything to Goneva's power through the heart of Northampton's defence.
Burns' conversion sent Leicester nine points in front, which put the ball firmly back into Northampton's court as the clock ticked down.
A 77th-minute Hanrahan penalty raised Northampton's hopes of a comeback, but Leicester had enough tactical nous to close out a deserved victory.