It secured their fourth straight league victory, the first time they have achieved the feat in the top flight since 1968, and saw them close to within two points of sixth-placed Arsenal.
That would be his best haul across one campaign for Tottenham and would also take Kane four goals behind Liverpool's Mohamed Salah, who is leading the way in the race to finish as the divison's leading marksman.
Mame Biram Diouf quickly cancelled out Christian Eriksen's opener for Tottenham on 52 minutes before the Danish playmaker scored again on 63 minutes to re-take the lead and ultimately win the match for Spurs.
"I know I'm up against a good side with a great manager behind them, but it's a game we can win".
Pochettino's fourth-placed side are only behind third-placed Liverpool on goal difference, with a game in hand on the Merseyside club, and crucially have an 11-point cushion on Chelsea in fifth.
Harry Kane claimed he did score Tottenham's victor at Stoke to break a personal goals record, even though Christian Eriksen has been credited with it.
"It was my goal, it flicked off my shoulder and went in, " he said.
Spurs won a sixth successive Premier League game at Stoke to strengthen grip on top four
Stoke level through Diouf, who scored a fortunate goal when Hugo Lloris' clearance slammed off his torso and into the Spurs goal.
Xherdan Shaqiri nearly secured Stoke a share of the spoils 10 minutes from time, but his free-kick came back off the crossbar and Spurs held on to the victory which leaves them only behind third-placed Liverpool on goal difference.
Pochettino was less concerned about the identity of the scorer.
But Spurs' Pochettino said: "I didn't have time to see it on TV, but Harry said that it touched him on the shoulder".
"The most important thing for me is the three points".
"If you ask me how proud I am of the team, I thought we were outstanding".
"We don't look like a team going down".