Hull had to initially overcome a 3-0 deficit after first period goals from Jade Portwood, Jay King and Loren Barron put the visitors in control.
But Omar Pacha’s side gained parity before the halfway point through second period strikes from Dominic Osman, Corey Tanaka and Eric Galbraith, only for the Capitals to forge ahead again through Greg Collins’s 29th-minute goal.
That was how it stayed until 56.54 when player/assistant coach Carl Lauzon levelled with his 30th goal of the season.
A goalless overtime led to the shoot-out, with Jordan Mayer and Lauzon on target to clinch the extra point.
With five games remaining, the result widened the gap on the Capitals to five points, while also taking Hull above Fife Flyers in the overall standings.
“It was a crucial win and we knew we needed the two points – we knew it was the most important game of the year,” said Pacha.
“After the first period I was frustrated and disappointed because there was a lack of execution on certain plays.
“But we battled through it, we got a good second and third period and we managed to win the game. We did it the hard way but we found a way to get the result.”
Stingrays return to action on Saturday when they welcome title-chasing Cardiff Devils (6.30pm) before heading to the West Midlands 24 hours later to take on form team Coventry Blaze.