Nuno Espirito Santo made eight changes from the team that lost to Arsenal in the north London derby at the weekend but Spurs got off to a fast start.
They had the lead inside four minutes as Dele Alli slotted home a penalty after being brought down by Mura goalkeeper Matko Obradovic.
Giovani Lo Celso made it 2-0 four minutes later with a neat finish from a tight angle, however, a stunning volley from Ziga Kous halved the home side's lead on 52 minutes.
Nuno then called on his star player to help get the job done just shy of the hour and Kane obliged in style with a 20-minute hat-trick.
Kane's treble made it a comfortable evening for Spurs, who claim their first win since August 29.
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