Robbie Brady put the home side's noses in front when beating England international Jordan Pickford from 20 yards out.
Chris Wood saw a huge chance go begging when he failed to beat the Everton stopper from Dwight McNeil's threaded pass.
And then the Toffees drew level on the stroke of half-time when Dominic Calvert-Lewin converted Richarlison's cross.
Both sides had chances to win it - as Nick Pope denied James and Gylfi Sigurdsson while Pickford kept out Wood's header - but the spoils were shared.
Here are the ratings...
1. Nick Pope 7
Possible question mark over whether he could have done more to cut out Richarlison's cross for the goal, but saved superbly from Calvert-Lewin, James and Sigurdsson, with the latter a real point-saver.
2. Matt Lowton 5
It didn't take long for Richarlison to unsettle the right back. The Brazilian forward got the better of Lowton throughout. Distribution also not at its best when playing out from the back.
3. James Tarkowski 6
Dealt with the movement and physicality of Calvert-Lewin for the most part, but Everton's tactical change opened things up ahead of him. Richarlison peeled away from him for the equaliser and the Toffees' goalscorer almost added a second when getting in behind.
4. Ben Mee 7
Magnificent in the air again and read the play ahead of him well, snapping into challenges when the ball was played into Calvert-Lewin. The centre back, however, didn't react quick enough to intercept Richarlison's cross for the leveller.