The turning point in the direction of the game was three two-pointers in a matter of seconds for Cheshire, who were breaking down Sheffield attacks at will.
They still trailed by 12 points but it had given them hope of a comeback, as both teams continued to swap points.
Four free throws from Hamilton in quick succession made the score 79-71 and when the game's dominant force batted the ball out of his team's danger-zone moments later, the Jets fans sensed something special was on.
The sense of expectation was increased when Sharks' Michael Tuck missed his free throw, before Hamilton pulled the score back to 81-78.
But Jets' Paul Peterson missed two free shots of his own with the score at 83-80 and Lawson's cool head made the difference as Sheffield held on for their win.
Marks scored 22 points, with Mike Cook leading the way for Sheffield's scorers with 19.
Sheffield player-coach Atiba Lyons told BBC Radio Sheffield:
"I feel great. I probably wasn't expected to win something so quickly in my tenure as head coach.
"A year and a half into it, it gives me more confidence as a coach. You're always unsure if you're doing the right things.
"Sheffield is a great basketball city. They support the team through thick and thin. I've been here for four years - we've had some good seasons and some bad seasons and you see the same faces.
"When you stay here long enough, you want to win it for them. They go to all the away games - Worthing, Plymouth. We don't even want to do it and we're playing. To win something for them, I'm speechless."
Cheshire Jets coach Paul Smith told BBC Radio Merseyside:
"I'm really frustrated. We were dominant in the fourth quarter - why did we leave it until then?
"I'm hoarse now from trying to get the guys going earlier than that.
"We have enough talent in this team to have won that game, possibly comfortably.
"But no disrespect to the Sharks. They deserve it and they won it as they played for four quarters."
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