Spurs dominated the first half and Gareth Bale put them ahead with a glancing header on a miserable Wednesday evening.
Bale was withdrawn at the break, leading to concerns he may have sustained another injury, and Jordan Thompson equalised with Stoke's first shot of the tie eight minutes into the second half.
Spurs were not to be denied a place in the last four, though, as Davies thumped in from distance before Kane sealed a first win in four games as Tottenham moved to within two victories of lifting their first trophy since 2008.
