The Barons never led Saturday.
OKC goaltender Yann Danis turned away the first 17 shots, but the Heat scored two goals in a span of 79 seconds in the second period.
By that time, Abbotsford had outshot the Barons 21-7.
"They were throwing everything at the net, crashing the net, and it paid off for them for two goals," said Danis, now 8-4 on the season.
OKC's Hunter Tremblay stopped the bleeding just 85 seconds later with a breakaway shorthanded goal to pull the Barons to within 2-1.
Chris VandeVelde tied the score at 2 with just 1:29 left in the second period on a put-back after rebounding Gilbert Brule's shot out front.
Nelson said he thought his team had weathered the storm and was ready to take control.
"I did. We had momentum," Nelson said. "It's too bad the period had to end."
Abbotsford took a 3-2 lead 3:25 into the third period and set the final margin with an empty net goal with 33.7 seconds left.
John Rohde: 475-3099. John Rohde can be heard Monday-Friday from 6-7 p.m. on The Sports Animal Network, including AM-640 and FM-98.1. Follow him on Twitter @RohdeOK.