Cenk Tosun's header narrowly missed the inside of the post and sliding team-mate Seamus Coleman, before Dominic Calvert-Lewin curled over in the final few minutes.
In the first half, Everton goalkeeper Jordan Pickford saved well from Dominic Solanke, partnering Danny Ings up front with Mohamed Salah injured and Roberto Firmino on the bench.
Liverpool's Loris Karius produced a spectacular save to keep out a curling effort by Yannick Bolasie.
After the break, substitute Alex Oxlade-Chamberlain fired over before Everton's late rally.
Everton and Liverpool remain ninth and third respectively after the draw.
Everton: 14 Apr v Swansea (A)
Liverpool: 14 Apr v AFC Bournemouth (H)