Higginbotham gave the Potters the lead in the 23rd minute, arriving at the far post to convert Matthew Etherington's corner.
Jones wrapped up the win with his third goal in three games, rising to meet Rory Delap's throw-in with 11 minutes remaining.
Stoke made the more positive start and after Tuncay Sanli had won a free-kick in a useful position but blazed it over, his strike partner Ricardo Fuller saw his header palmed around the post by Mark Schwarzer.
Tempers flared just before the quarter-hour mark when Etherington brought down Moussa Dembele with a rash challenge.
Cottagers captain Danny Murphy reacted by throwing the ball at Etherington, and after a minor skirmish, referee Mark Halsey booked both players.
Murphy then went some way to redeeming himself by blocking a shot from Jones in the box, before Brede Hangeland sent a header goalwards at the other end but could not trouble goalkeeper Asmir Begovic, who was making his first appearance of the season having been left out of the Potters squad to face Shrewsbury in the previous round after he reportedly refused to play.
Moments later Stoke went in front thanks the head of Higginbotham, who made the perfect connection with Etherington's corner by crashing the ball across Schwarzer and in.
The hosts were good value for their lead with Tuncay, in the team for Jon Walters, looking particularly lively.
The Turkey international fired an acrobatic effort from outside the box which Schwarzer was equal to and Etherington also tried something spectacular soon after, slipping as he hit a shot from distance that went over the crossbar.
Fulham won the ball back and Clint Dempsey teed up Zoltan Gera, who drove powerfully at Begovic.
Dempsey then saw his free-kick deflect over off the Stoke wall and scuffed a shot into Begovic's arms.
Jones almost scored within seconds of the restart as Stoke surged forward, but the Trinidadian's effort flew wide.
Play switched and Carlos Salcido sent an effort off-target before the hosts survived a real scare.
Begovic flapped at a cross from Gera in the 54th minute and the ball dropped to Dempsey, but the goalkeeper's blushes were spared as the American volleyed it over the crossbar.
Fulham boss Mark Hughes responded by introducing another attacking option, with Eddie Johnson coming on for Simon Davies.
It was Stoke who came closest next, Glenn Whelan receiving Tuncay's free-kick and unleashing a drive that whistled just over.
The Potters were sloppy in the box again soon after though, with Dempsey almost capitalising but unable to guide the ball beyond Begovic.
Fulham were suddenly looking the more incisive outfit and Stoke did just about enough to put Johnson off as he fired an effort wide.
Dembele then sent a fierce effort the wrong side of the post and was involved again moments later in a move that resulted in Dickson Etuhu heading over.
By this stage though, the Potters had Delap, a 68th-minute replacement for Salif Diao, on the field, and it did not take long for him to make the desired impact.
A little over 10 minutes after coming on, Delap launched a trademark long throw into the Fulham box and Jones beat Schwarzer to head it home.
Gera curled a shot wide and Stoke fans cried out for a penalty when Andy Wilkinson was tackled late on.
Wilkinson then picked up a booking after bringing down Dembele, who had to be carried off on a stretcher.
Substitute Danny Pugh had a shot saved late on, but the hosts had done enough and their place in last 16 was confirmed as Halsey blew for full-time.