The Blues shaded a first half that featured few genuine chances, Luke Garbutt driving a shot wide and Harry Charsley forcing a save from goalkeeper James Talbot.
And David Unsworth's men remained in the ascendency after half time, with David Henen shooting over and Jose Baxter missing the target before Bassala Sambou missed the best chance thus far from four yards out.
Shane Lavery went close later in the second half but with five minutes to go, Luke Molyneux fired a solo effort past Joel Robles to score what would prove to be the winner.
Morgan Feeney had a couple of late headers that both missed the target as Everton searched for an equaliser that never came.
Everton U23s: Robles, Charsley, Garbutt, Baningime, Feeney (c), Foulds, Evans, Adeniran (34' Donkor), Sambou (70' Lavery), Henen, Baxter (70' Broadhead).
Subs not used: Gray, Markelo.