It was also Stoke’s first home loss since January as Spurs maintained their push for a Europa League place.
The only goal of the game arrived just after the half-hour mark with Rose scoring his first league goal since his debut strike against Arsenal more than four years ago.
It was in no small part down to Emmanuel Adebayor who danced around Glenn Whelan before chipping it back for Rose to head home from eight yards.
The second half was marred by the sending-off of Stoke captain Ryan Shawcross, booked for a foul on Kane late in the second period – he left his foot in on Rose and left Andre Marriner with little option but to dismiss him.
Stoke players and fans were furious, but there was little doubt Marriner got the decision right.
With ten men Stoke pushed on and they could and should have levelled with Steven Nzonzi, Peter Odemwingie and, particularly Marko Arnautovic, all wasting good chances to level as Spurs held on for the win.
The victory takes Spurs to within three points of Everton in fifth, whilst Stoke stay in 10th.
The visitors were quicker out of the blocks and Nacer Chadli shot wide after only 19 seconds, while Asmir Begovic had to react quickly to push Aaron Lennon’s mis-hit cross past the post.
Shawcross was furious with what he felt was a forearm to the face from Adebayor but Marriner took no action.
The home side withstood the early pressure to enjoy a decent spell themselves, and Tottenham were grateful for the presence in their defence of captain Michael Dawson, back after a hamstring injury.
The centre-back was key in dealing with the threat of former Spurs striker Peter Crouch, who was making a nuisance of himself in the penalty area.
Crouch was involved as Stoke came close to taking the lead in the 27th minute, playing in Stephen Ireland, who cut in from the left and unleashed a powerful shot that just curled beyond the far post.
But six minutes later it was Tottenham who opened the scoring when Adebayor beat Whelan on the right of the area before sending in a perfect cross for Rose, who planted a fine header past Begovic.
And the visitors might have extended their lead just before half-time when Harry Kane, in search of his fourth goal in four games, was given time and space but he dwelled on the ball a fraction too long and his shot was blocked.
Stoke began the second half brightly with a shot from Arnautovic that flew wide but their hopes of getting anything from the game were badly dented in the 52nd minute when Shawcross was shown a second yellow card.
The defender, who had been booked for a foul on Kane in the first half, caught Rose on the shin with his studs.
Referee Marriner initially did not even give a free-kick but, with Rose still on the floor, pulled back play and produced the red card, much to the anger of the home fans.
Rose was being booed every time he touched the ball and the full-back was booked for pushing Geoff Cameron in the chest after the Stoke man had felled him.
His game was over soon after as Sherwood took him out of the firing line to be replaced with Zeki Fryers.
Stoke were fired up and were enjoying their best spell of the game, with Peter Odemwingie, Arnautovic and Steven Nzonzi all missing the target.
Arnautovic then produced a much better effort that was rather awkwardly blocked by Hugo Lloris.
At the other end, Eriksen sent a free-kick just over the bar, but it was Stoke doing most of the attacking and their calls for a penalty were in vain as Dawson and Odemwingie clashed.
Their clearest opening came in the 84th minute as Arnautovic broke through but the Austrian shot tamely at Lloris, and Begovic had to be at his best to deny Paulinho in stoppage time.