The left-handed pair had already settled any nerves by the time 29-year old Ervine was caught and bowled by Krishna Chandran with just 36 runs from seven overs required.
Zimbabwe captain Elton Chigumbura came in to see his side safely to their target of 286 with two overs to spare and he finished on 14 not out.
UAE, making their first World Cup appearance since 1996, posted a competitive 285 for seven and then took wickets at crucial moments with wicketkeeper Brendan Taylor's dismissal in the 28th over looking almost fatal at the time.
Taylor was well set and looking to up his scoring rate when he attempted to sweep a full delivery from Nasir Aziz and was struck low on his front pad.
He was given out for 47 but reviewed the decision only for technology to show the ball would have hit the middle stump.
Their tails up, UAE looked on course to record their first win over a test nation in their 20th official one-day international until Williams and Ervine intervened.
The victory ended Zimbabwe's losing streak of seven matches stretching back to their three-wicket win over Australia in Harare last August.
UAE's innings had been well constructed with all but two of their batsmen getting past 20 and Shaiman Anwar stroking his highest score in one-day international cricket.
Anwar scored 67 and was ably supported in an 82-run partnership by wicketkeeper Swapnil Patil (32), while Khurram Khan (45) and Krishna Chandran (34) also produced an 82-run third-wicket partnership.
Mohammad Naveed (23) and Amjad Javed (25) combined for a quick-fire unbroken stand of 53 at the end of the innings to drive the UAE to their highest score in a one-day international.
Tendai Chatara finished with figures of 3-42.
(Reporting by Greg Stutchbury in Wellington; Editing by Nick Mulvenney)
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