Coles had appeared to be have fallen short and then made a double movement to get the ball over the line in the 85th minute, but television official Mike Fraser ruled he had scored the try.
Fly-half Beauden Barrett then converted to give the home side their slim victory and ensured they stayed in the hunt for the play-offs, moving to 57 points on the table, though they need other results to go their way to make the top six.
The Chiefs (64) had already secured top spot in the New Zealand conference and at least second overall for the play-offs.
The home side were placed under early pressure from the Chiefs, leaking a try to scrum-half Tawera Kerr-Barlow that was converted by Aaron Cruden, but then had the better of the next 25 minutes with winger Julian Savea scoring two tries, both times finishing off superb build-ups by the entire team.
Tighthead prop Ben May, however, was then given a yellow card by referee Jonathan Kaplan for a dangerous tackle on Sonny Bill Williams, and the Chiefs used the extra-man advantage to level the score with a Cruden penalty and Ben Tameifuna try.
Barrett, who had earlier missed two penalties and a conversion, gave the home side an 18-15 half time lead when he slotted a penalty after the hooter had sounded.
The Chiefs increased the pressure in the second half, virtually camping down in Hurricanes territory, with the home side's only real attack inside the first 20 minutes after the break leading to Cruden's try when he intercepted a Barrett pass intended for Savea.
Cruden's try forced the Hurricanes to chase the game, making numerous mistakes as they tried to force the play rather than be as composed as they were in the first half, while the Chiefs slowed the game down.
The Hurricanes, however, somehow managed to get down field in the final minutes and recycle the ball continuously before Coles scored the match winner to bring the 23,549-strong crowd to their feet.