The Blues were trailing at the Landor Training Centre to Jordon Garrick's heavily-deflected shot when Luke Garbutt struck from the penalty spot to level the game.
The contest looked to be heading to a 1-1 draw as Brandon Cooper was sent off for a foul on Jose Baxter and the Everton man was forced off with an injury that also reduced the visitors to 10 men as Jeffers had used all his substitutes.
But Fraser Hornby found Lavery with a cross and the forward finished emphatically in stoppage time to hand the Blues the points.
Scorers: Garbutt (74'), Lavery (91')
Everton U23s: Hewelt, Charsley, Garbutt, Adeniran, Feeney, Foulds, Evans, Baxter [Y], Sambou (72' Hornby), Broadhead (71' Lavery), Bowler (76' Donkor).
Subs not Used: Hilton, Markelo.