Michael Beale’s team were largely comfortable throughout the first hour, and took the lead through a controversial penalty converted by Harry Wilson.
The Reds then increased their advantage again through Joao Teixeira and Sergi Canos, before Joe Williams and Conor McAleny scored goals either side of half time to make for a nervy finish.
And with just over five minutes remaining, Everton scored their third as Davies capitalised on a goalkeeping error.
Everton U21 boss David Unsworth also handed a debut to new signing Mason Holgate, a week after completing a £1 million move from Barnsley.
The game got off to an even start, and the opening foray lacked any real spark. But as the half wore on the visitors started to grow into the game and began to threaten.
And on ten minutes they got their chance to go ahead, courtesy of a dubious penalty decision which saw Holgate and Wilson come together what looked to be outside the area initially
But referee Scott Oldham had no doubts about it, pointed to the spot, and Wilson stepped up to calmly stroke the ball to the goalkeeper’s right from 12 yards.
The Reds were soon two to the good after Everton got caught in possession. The ball found its way to the feet of Portuguese forward Joao Teixeira and his driven effort from outside the area found its way past Hewelt to make it 2-0.
The hosts were continually frustrated throughout the opening 45 minutes, and were unable to truly test Ryan Fulton in the Liverpool goal.
It could have been 3-0 sooner than it was, as Dunn headed against the post after Hewelt could only get fingertips to Jordan Rossiter’s troubling cross.
Though Liverpool continued to showcase their endeavour, and Sergi Canos lashed the ball home after the young Blues were unable to deal with the offensive pressure.
Williams’ effort – which looped over Fulton off the head of defender Dan Cleary – offered Everton some hope heading into the second half.
They looked to capitalise on the late spark, and McAleny manage to halve the two-goal deficit with an almost carbon copy of Teixeira’s earlier goal.
On the front foot for the majority of the second period, Everton troubled, and came close twice thanks to balls from Gethin Jones – but neither of his targets could convert their promising chances.
However, they found an unlikely equaliser through Davies, who turned in the rebound after Fulton couldn’t handle Matthew Pennington’s shot.
The draw extends Everton’s mini-derby victory drought – which stretches all the way back to May 2011, when goals from Eric Dier – now of Tottenham Hotspur – Jose Baxter and Hope Akpan gave The Toffees a 4-1 away win.
The point does mean that both sides get off the mark in the U21 Premier League, meaning that all 12 teams now have at least one point.
Everton U21: Hewelt (GK), Jones, Connolly, Walsh, Pennington, Holgate, Grant (Robinson, 46), Williams (C), Byrne, Evans (Davies, 46’), McAleny Subs: Stanek (GK), Ledson, Davies , Lavery, Robinson
Scorers: Grant (46’), McAleny (65’), Davies (84’)
Bookings: Walsh (49’), Davies (52’)
Liverpool U21: Fulton (GK), Randall, Maguire, Cleary, Llori (Brewitt, 46’), Rossiter, Canos (Chirivella, 62’), Brannagan (C), Dunn, Teixeira, Wilson Subs: Firth (GK), Chirivella, Kent, Brewitt, O’Hanlon
Scorers: Wilson (13’), Teixeira (22’)
Referee: Scott Oldham (Lancs)