Astley Arms halved the deficit early in the second half following a misunderstanding in the Manchester penalty area, but Louie Basso restored the two-goal advantage after a sliderule through ball from Alex Singer.
Goals in the last 15 minutes from Stevens and Max Rose secured the points.
Manchester joint-manager Stuart Delmonte said: "We played some wonderful one-touch football and they just couldn't live with us.
"The Astley manager told me after the game that we’re by far the best team they’ve played over the last two years."
Maccabi host Heartshead in the fifth round of the Brian Brockbank Trophy next weekend.