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United were good value for the win over the Stoke side, in which they kept their third clean sheet of the season.
He said: "We defended resolutely, which we have to do in this division, and that gave us the platform to attack.
"We need to keep our feet on the floor, but realise that we have a genuine chance of promotion."
Richard Hinds opened the scoring with a low drive from the edge of the area after 32 minutes.
And Andy Keogh doubled the advantage five minutes later when he headed home Beagrie's cross-cum-shot.
Beagrie added: "We dominated and played expansive football in the first-half, we put them under intense pressure.
"We need to really kill sides off. I missed a good chance after the break that would have put a different reflection on the game.
"I should have scored, and that's disappointing. But overall we totally deserved the win."