The hosts went 2-0 up after the left back’s shot had been adjudged by the linesman to have crossed the line amidst a goalmouth scramble in which Hammers goalkeeper Raphael Spiegel appeared to have made a magnificent save.
But, by and large, the Blues took charge in the second period, with Conor McAleny’s brilliant strike from outside the box breaking the deadlock just five minutes after the restart.
West Ham, featuring Ravel Morrison who played after being cleared of assault charges, were unable to offer little in the way of a reply, and they slumped to their second consecutive defeat.
West Ham started the game the brighter of the two sides, but they were unable to capitalise on any of the three corners they had in the opening four minutes.
The best chance of the opening 45 came when Connolly went on a marauding run into the visitors’ box, but his powerful strike cannoned off the underside of the crossbar and bounced out to safety.
The hosts came close once more, but McAleny was unable to get his shot on target as he fell to the ground under defensive pressure.
But one of the home side’s best opportunities to put themselves in front fell to Courtney Duffus, but he was unable to steer his header goalward from Gethin Jones’ great cross.
An uneventful and dour first half came to a close, but the interval enabled both sides to rethink before emerging for the second period.
David Unsworth’s half time team talk appeared to work well, as Everton improved after the break and were rewarded just five minutes in.
McAleny picked up Grant’s pass and unleashed a wicked shot from 25 yards which the scrambling Spiegel was unable to keep out.
And Everton were soon two to the good, when Grant’s cross wreaked havoc in the West Ham penalty area, allowing Connolly to bundle the ball towards goal.
At first sight the goalkeeper appeared to have palmed the ball out of danger, but the linesman on the far side signalled that the ball had crossed the line, and the visitors’ appeals were duly waved away by referee Paul Graham.
Everton continued to push forward, and a jinking Matthew Kennedy run saw him beat several West Ham defenders before forcing a save out of Spiegel.
The Londoners struggled to break Everton down, and their evening of frustration was summed up when Dan Potts headed a Sam Westley cross well wide.
Chris Long, who had earlier replaced goalscorer McAleny, was inches away from sealing the points with a curling effort from outside the box, but Everton were able to hold on.
Everton U21 (4-1-4-1): Hewelt, Jones, Connolly, Júnior, Browning, Kenny, Kennedy, Grant, Duffus, McAleny (Long 60’), Henen Subs: Hunt, Dowell, Ledson, Williams, Long
West Ham United U21 (5-3-2): Spiegel, Westley, Page, Henry (Oxford, 65’), Burke, Potts, Bywater (Cullen , 65’), Poyet, Lee (Turgott, 65’), Morrison, Nasha Subs: Oxford, Boness, Matthias, Cullen, Turgott
Referee: Paul Graham