The all-Berkshire derby ended 4-4 before both sides missed all three of their penalty shots taking the league fixture into sudden death.
Forward Dan Davies scored his penalty to make himself a Jets hero after Rob Lamey missed for Bracknell.
Ryan Watt put Slough ahead in the fourth minute before Peter Jasik equalised for the Bees on 13.48.
However, two goals inside 60 seconds from Adam Calder and Darius Pliskauskas saw Slough end the first period 3-1 ahead.
Bracknell's Carl Graham scored the only goal of the second period to halve the lead.
Lukas Smital levelled the scores less than a minute into the final period before Calder restored the Jets lead with his second of the game.
But, James Galazzi's late strike forced the derby into overtime.
The extra five-minute period, played four-on-four, could not seperate the sides so penalty shots followed.
Alex Barker, Shaun Thompson and Matt Foord all missed for the Bees while Pliskauskas, Calder and Connolly failed to score for Slough.
However, Davies secured the points for the visitors when netting against Bracknell stopper Carl Ambers.
"It was a tough weekend but the guys battled hard and we found a way to win," head coach Sheppard told The Observer.
"I'm really happy with the four-point weekend.
"You never want to give up a lead, but Rocky (Greg Rockman, net minder) bailed us out.
"Dan Davies has come up huge for us in overtime all year and he did it again on Sunday so it was a great result.
"Defenceman Sam Waller has also been great, he made a massive play with four seconds left to keep the puck out and send the game to the shootout."
Sheppard continued: "It's a tough place to come, they've had our number from pre-season so we're definitely happy to get the two points. It's the first time we've beaten them here and they've got a great team.
"We just keep rolling and finding a way to win, it's definitely a great start to the
season for us."
The win extends Slough's lead at the top of the English Premier Ice Hockey League to three points from the Sheffield Steeldogs.
Sheppard's Jets welcome Basingstoke Bison to The Hangar on Saturday (6:30pm f/o) and travel to Guildford Flames on Sunday (6pm f/o). Both fixtures are for league and cup points.
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