The outcome sent group leaders South Africa, who have six points from three games, into the Super Eight alongside defending champions India and two-time winners West Indies.
New Zealand, with four points ahead of their final group match against bottom-placed Canada on Tuesday, look primed to claim the second Super Eight spot from Group D.
Sohaib Khan (68) led a spirited recovery to power UAE to a respectable 160-9 after being put in to bat.
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. We just need to come up with a positive mindset.
"Even if we don't make it to the next round, we want to play with the same brand of cricket and make our nation proud."
Earlier, TV umpire Paul Reiffel found himself in the thick of action in the first four overs after Rashid won the toss and elected to field at Arun Jaitley Stadium.
UAE opener Aryansh Sharma edged the first ball from Omarzai but it was unclear if wicketkeeper Rahmanullah Gurbaz had taken a clean catch. Reiffel went through umpteen replays before giving the benefit of the doubt to the batter.
Three balls later, Reiffel was tasked with a similar job and this time the official decided that Aryansh was indeed out, caught behind for a duck.
Off-spinner Mujeeb Ur Rahman trapped UAE captain Muhammad Waseem lbw for 10, a decision which Reiffel upheld after the batter reviewed it.
In the fourth over, Sohaib was initially given lbw to Mujeeb but he survived after replays confirmed the ball would have narrowly missed the leg stump.
The reprieve allowed Sohaib to combine with Alishan Sharafu (40) in an 84-run stand to steady the UAE 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.