Striker Harrison capped a fine team performance in the first period when he glanced home Nathan Oduwa's cross to cancel out Mason Mount's early free-kick, which had seen the Blues edge themselves in front.
Dominic Solanke restored the visitors' lead 10 minutes into the second period on a chilly afternoon at Hotspur Way but three minutes later, left-back Connor Ogilvie popped up with a second equaliser from a corner.
Having controllled the play for much of the game, the last 20 minutes saw both sides go close from set pieces while Pau Lopez made an important low save, but the points were ultimately shared.