The outcome sent group D leaders South Africa, who have six points from three games, into the Super Eight joining defending champions India and two-time winners West Indies.
Despite Afghanistan's efforts New Zealand look primed to claim the second Super Eight spot from Group D. A win in their final group match against bottom-placed Canada on Tuesday will send them through.
Sohaib Khan (68) led a spirited recovery to power UAE to a respectable 9-160 after being put in to bat on Monday.
Omarzai, who claimed 4-15 with the ball, returned to hammer 40 not out off 21 balls to secure 2024 semi-finalists Afghanistan's first win in this year's tournament with four balls to spare.
"Always important to have a win at the World Cup and let's hope for the best in the rest of the competition," Afghanistan captain Rashid Khan said.
"We still have a little bit of hope that we can make it to the next stage."
TV umpire Paul Reiffel found himself in the thick of action from the start giving a reprieve to UAE opener Aryansh Sharma first ball from Omarzai after umpteen replays to check a clean catch behind, but then adjudging him out the same way for a duck three balls later.
After off-spinner Mujeeb Ur Rahman trapped UAE captain Muhammad Waseem lbw for 10 Sohaib combined with Alishan Sharafu (40) in an 84-run stand to steady the innings.
Omarzai ended Sohaib's fine knock, which included four sixes and six fours.
Afghanistan lost their in-form opener Gurbaz in the first over but first Ibrahim Zadran (53) and then Darwish Rasooli (33) kept them in the hunt.
Omarzai finally sealed Afghanistan's win with a four off Junaid Siddique.
