The Potters hit the post through Delap and saw their own crossbar rattled by John Arne Riise before Walters broke the deadlock in the 80th minute, diverting Matthew Etherington’s effort home.
Delap then doubled their advantage seven minutes later, nodding in an Etherington free-kick.
Stoke’s deserved win continued their solid start to the campaign, Tony Pulis’ side having now taken 12 points from their opening eight Premier League fixtures to consolidate their position in the top half of the table.
Pulis made two changes from the defeat at Swansea last time out, with Etherington, having been given a sugar injection for a back complaint, returning to the Potters team along with defender Matthew Upson.
Striker Kenwyne Jones, who had been sidelined for three matches by a hamstring injury, was among the substitutes, while the visitors’ first XI showed one adjustment from their 6-0 thumping of QPR, Aaron Hughes coming in for Zdenek Grygera after recovering from a groin problem.
Fulham won a corner in the fifth minute and Danny Murphy’s delivery was flapped at by Stoke goalkeeper Asmir Begovic, but also missed by everyone else.
Murphy sent a free-kick wide of the target before a well-worked set-piece move by the hosts saw the ball fall to Andy Wilkinson, who put it into the side-netting.
Marc Wilson then made a crucial intervention to prevent Mousa Dembele’s through-ball finding Bobby Zamora and Riise tried his luck from outside the area, but his shot whistled over the bar.
Peter Crouch attempted to volley a smart cross from Jermaine Pennant but failed to make proper contact.
Six minutes later the ball came to Crouch in the box again, Wilson’s effort ricocheting off the Fulham wall, but the tall striker saw his shot blocked.
After a sluggish start Stoke were beginning to exert some pressure and Cottagers keeper Mark Schwarzer had to be alert to beat Crouch to Etherington’s cross.
Dembele, who had picked up an injury, came off for Damien Duff before Etherington linked up with Walters, only to roll a tame shot wide.
Five minutes before the break Zamora nearly put Fulham in front with a header that went just the wrong side of the post.
Pennant then came similarly close in stoppage time, playing a one-two with Crouch but dinking the ball narrowly wide.
Shortly after the restart, Delap lashed a shot from distance that was rising too fast to trouble Schwarzer.
Delap then caused a brief scramble in the box with one of his trademark long throw-ins that Fulham managed to hack clear, before the midfielder picked up the first booking of the game for a challenge on Clint Dempsey.
Schwarzer made a hash of dealing with another Delap delivery in the 64th minute, the Australian twice failing to claim the ball and the visitors eventually surviving thanks to a goal-line clearance as Crouch tried to force it home.
It was proving an eventful time for Delap, who then sent a deflected drive crashing against the post.
Stoke continued to press, with Wilson curling a free-kick over the bar.
The Potters were then denied on the line again, Murphy getting in the way of Wilson’s header and referee Martin Atkinson waving away protests that the Fulham captain had handled.
For all their momentum, the hosts almost fell behind in the 78th minute as Riise’s fiercely-struck free-kick crashed back off the bar.
But two minutes later Stoke grabbed the lead as Walters stabbed in Etherington’s shot.
Delap then wrapped up the points with three minutes of normal time remaining, rising to meet Etherington’s free-kick from the left and head it past Schwarzer.