The visitors were a little slower off the mark in the second half but Reece James stole possession from Ben McLoughlin to force a diving stop out of Stringthorpe and Tyias Browning's risky lunging tackle on Will Keane, as he charged onto a Guillermo Varela through ball, paid off.
Anderson was withdrawn just past the hour and Everton were certainly enjoying more of the game with Fabio's attacking threat largely contained. Chris Long and Joe Williams looped volleys wide and Hope capitalised on a slip by Varela to meet a Long pass with a goal seeming certain until Amos superbly diverted the shot onto the left-hand post.
It was a vital save and provided a springboard for some overdue United attacks as Will Keane tested Stringthorpe again following a Lawrence run before the Welshman was replaced by James Weir. The woodwork came to the rescue again when Long robbed Michael Keane and belted a low drive beyond Amos' drive as the outcome remained in the balance entering the final 10 minutes.
There was to be no late drama with both sides looking a little tired as the score remained deadlocked at 0-0 on a wet night in Merseyside.
United: Amos; Varela, Thorpe, M.Keane, Fabio; Lawrence (Weir 76), Ekangamene, Anderson (Blackett 62), Cole, James; W.Keane. Subs not used: Barmby, Sutherland, Love.