England were 221 for five in reply to West Indies' first innings 426, a deficit of 205 runs and needing a further 56 to avoid the follow-on, in a match where the first two days had already been washed out completely.
Ian Bell was 76 not out and nightwatchman Steven Finn nought not out.
West Indies' total owed much to Tino Best's whirlwind 95, the highest score ever by a Test match No 11 batsman.
Best faced just 112 balls with a six and 14 fours on Sunday's fourth day.
Together with wicket-keeper Denesh Ramdin, who made 107 not out, Best shared a partnership of 143 -- the third highest 10th wicket stand in Test history and a West Indies record.
And, before Sunday's close, fast bowler Best removed both England captain Andrew Strauss and Jonathan Bairstow.
England, with the series already won at 2-0 in their favour, controversially chose to rest first-choice new-ball pairing James Anderson and Stuart Broad from this match.
In their places came fellow seamers Graham Onions and Finn.
Durham quick Onions, who two years ago suffered a career-threatening back injury, took four wickets for 88 runs in 29.3 overs, with Finn managing three for 109 in 32.
After this match, England and the West Indies play three one-day internationals and a stand alone Twenty20, starting with Saturday's 50-over clash at Southampton.
The West Indies will warm-up for the limited overs fixtures with a one-day game against Middlesex at Lord's on Wednesday.