Spurs had led through a 25th-minute Lucas Moura goal and Newcastle had not managed a shot on target but deep into stoppage time Newcastle's Andy Carroll and Eric Dier challenged for a ball and the forward's header brushed the arm of the Spurs defender.
A VAR review led referee Peter Bankes to look at the incident, in which Dier had his back to the ball as he jumped with Carroll, and point to the spot.
Wilson's spot-kick squeezed past Hugo Lloris and earned Steve Bruce's side a point that had looked unlikely for most of the match.
Darlow also did well to save a Kane header but Spurs broke through in the 25th minute -- Son Heung-Min fed Kane on the left and his low ball across the face of goal was turned in by Moura at the far post.
Son fired a left-foot shot against the post before striking the woodwork again with a drive against the bar three minutes before the interval.
Mourinho took Son off at the break, perhaps taking into account the team's busy schedule with midweek League Cup and Europa League matches on the horizon, but without the South Korean there was a more pedestrian look to Tottenham.
Still, Mourinho's side appeared to have done enough until the latest handball drama which caused the Portuguese manager to storm down the tunnel.