Piquionne, without a goal in the Premier League, struck early in each half to keep the 2008 FA Cup winners on course for a return to Wembley - and keep under-fire Hart in a job.
Portsmouth made six changes from Saturday's dour goalless draw at Hull, while Stoke changed all ten outfield players.
The home side should have taken the lead after just nine minutes when Frenchman Frederic Piquionne came marauding down the right before crossing for Hassan Yebda, but the Algerian could only hook his shot into the air from six yards.
Piquionne showed Yebda how it was done eight minutes later, ghosting in at the back post to nod Nadir Belhadj's inswinging cross beyond a helpless Steve Simonsen with a towering header.
Stoke failed to force Portsmouth goalkeeper Jamie Ashdown into a meaningful save during a one-sided first half and Piquionne should have doubled his and the hosts' lead before the interval.
Yebda beat Danny Higginbotham down the Stoke left before pulling the ball back for Piquionne, but the on-loan Lyon striker blazed high over Simonsen's crossbar.
Portsmouth's passage into the last eight was secured by two goals in four minutes early in the second half.
Piquionne turned provider in the 55th minute, picking the ball up down the left before cutting inside Andy Griffin and sliding the ball through for Danny Webber to finish neatly from just inside the penalty area.
Moments later man of the match Piquionne capitalised on some comical Stoke defending to claim his second of the evening.
Simonsen collided with Higginbotham in trying to deal with Michael Brown's free-kick and Piquionne had the simple task of side footing into an empty net from two yards.
Piquionne almost completed a deserved hat-trick when he burst into the penalty area only to shoot into the body of Simonsen from a tight angle.
Substitute Kanu compounded Stoke's misery nine minutes from time, racing on to Piquionne's through ball before dancing his way round an embarrassed Simonsen and walking the ball into an unguarded net.