The Cobblers – who dumped Liverpool out in the second round – came into the clash at Portman Road languishing in 91st place in the Football League ladder yet looked set for another shock when Liam Davis put them ahead early on.
But goals from Carlos Edwards and Irish defender Damien Delaney soon had the Championship side in front and Keane’s men turned in a professional second-half display to seal their progress to the last eight with Tamas Priskin adding a late third.
Elsewhere, Birmingham are through to the quarter-finals of the Carling Cup after overcoming League One Brentford 4-3 on penalties.
Blues keeper Maik Taylor made the decisive save to deny Craig Woodman from the spot.
The Bees looked to have followed up their victory over Everton with success at St Andrew’s through Sam Wood’s 15 yard drive mid-way through the second half.
But veteran striker Kevin Phillips broke their hearts in the 92nd minute.
Phillips and Gardner both had chances for Birmingham in extra-time but the game went to penalties and Phillips, Gardner, Scott Dann and Carr scored from the spot for the home side, with Bowyer shooting over the bar.
Kevin O’Connor, David Hunt and Lewis Grabban netted for the Bees but Spillane hit the bar, before Taylor denied Woodman with the final kick.
Meanwhile, Mauro Boselli finally started to pay off his £6 million (€6.8m) transfer fee as Wigan earned their place in the quarter-final draw after a 2-0 defeat of Swansea.
The Argentinian striker claimed his first goal since his summer move from Estudiantes as Wigan manager Roberto Martinez secured victory over his former club at the DW Stadium.
Cedric van der Gun should have given the npower Championship visitors an early lead but Boselli gave Wigan a 51st-minute advantage and Ben Watson added a late penalty.
Leicester goalkeeper Conrad Logan gifted West Brom their place in the hat with a clanger that swung the game firmly in the visitors’ direction.
The sides were locked at 1-1 apiece after a Simon Cox strike for Albion and a Nicky Shorey own-goal, when Logan spilled the ball at Somen Tchoyi’s feet to leave the hosts behind again.
Steven Reid and Cox, with a second, added the gloss on a 4-1 win.