Borough had by far the better of the first half, Mitchell Cole firing narrowly wide early on.
Sills forced a smart save out of Clarke and then hit the woodwork after Michael Bostwick fired wide.
Deering had Oxford's best first-half effort but his shot from 25 yards was saved by Chris Day.
Stevenage finally broke the deadlock minutes after the restart when Bridges was fouled in the box, setting up Laird to net the winner.
Clarke kept Oxford in it when he tipped Beardsley's shot around the post.
And although Oxford improved after falling behind they could not find an equaliser, leaving their own title hopes on a knife edge.
Day tipped Clist's free-kick over the bar, while Matthew Green and Deering both failed to hit the target.
Stevenage manager Graham Westley said: "No-one is going to play down the significance of beating another contender, because obviously it makes a big difference to the league table, but it is still just another three points and we still have a lot to do.
"I thought it was an authoritative performance.
"We were nice and solid, we dictated play, and we were probably unfortunate not to go in ahead at half-time. But we came out for the second half with a real sense of purpose."
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