Rashid, who had the worst bowling figures on debut in the first innings of 0-163, had an unbelievable change in his fortunes as he grabbed 5-64 to dismiss Pakistan for 173 on a weary fifth day Sheik Zayed Stadium pitch.
England tried hard to chase down the 99-run target in a possible 19 overs but umpires called off play eight overs early due to bad light with the visitors on 74-4.
A result had looked highly unlikely on the fifth morning as England, replying to Pakistan's 523-8 declared, closed their first innings at 598-9.
But Pakistan's notoriously independable batting changed the complexion as Rashid exploited the rough with seven wickets falling for 71 runs, the last five for only 14 in the post tea session.
Only skipper Misbah-ul-Haq put up some resistance with 51 on a day when 15 wickets fell, compared to 16 over the first four days.
England lost Butler (4), Ali (11), Ben Stokes (two) and Jonny Bairstow (15) in the 11 overs.
It was a lucky escape for Pakistan as there have been 14 previous instances when teams had lost a Test after scoring 500 points or more, the highest by Australia when they lost to England after scoring 586 in Sydney in 1894.
Despite a slow and flat pitch both teams will regret missed opportunities which allowed Malik (caught off a no-ball on 40) to hit 245 in his comeback match and Cook scored 263 (dropped on 147 and 173).
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