The result saw Andy Adamson’s side leapfrog Sports into third position in the league table, where they sit just a point behind second-placed Goal Sports.
Matty Whitehall opened the scoring in the first half when he got forward from centre-half to curl a 25-yard effort into the top corner of the visitors’ net.
There were no more goals before the interval, but Eastway scored on the counter-attack to level the scores 10 minutes after the resumption.
Despite plenty of pressure from Sherburn, the visitors remained on level terms until the 75th minute when Whitehall nodded home a corner to claim his second of the night.
The points were wrapped up just a few minutes later when Paul Mills set up Gary Tyson to fire in number three from close range.
“We’re still in the hunt for promotion, and I’m pleased with how we’re doing,” boss Adamson said.
“We have Goal Sports to play twice which will go a long way to deciding who goes up.
“It’s been tough having to play so many games in such a short period of time recently, but we’ve won three games in the space of five days, which is a great effort.”
Two-goal Whitehall was named as the village side’s man of the match.