The hosts were quick out of the starting blocks and tested Ben Foster early on through Jan Vertonghen and Kane, before the England striker opened the scoring 12 minutes in after collecting Christian Eriksen's through-ball and lashing home via the inside of the post.
West Brom found themselves under siege as they withstood a succession of corners and saw their goal threatened by the likes of Danny Rose, Kyle Walker and Kane, who could have easily completed his hat-trick before the half-time whistle.
Tony Pulis's charges could count themselves lucky to only be two goals down by the break, but despite a brief revival early in the second half, it was soon business as usual as Spurs came at their opponents with increasing determination and ferocity.
The visitors' second-half resistance gave way in the 77th minute when Kane added his second with a fine volley, and just five minutes later he made it four by rifling home from Alli's skilful scooped pass.
The result sees Spurs leapfrog Liverpool into second place, while West Brom remain eighth in the table.