The hosts made 133-7 in their innings in Abu Dhabi, with Boyd Rankin taking 3-17 and Max Sorensen 2-16.
An opening stand of 61 between William Porterfield (72) and Paul Stirling (28) put Ireland in control and they appeared on course for victory at 91-1.
Eight wickets fell for 37 runs as the UAE avenged Sunday's 34-run defeat.
SP Patil top-scored for the UAE with 31 off 37 balls, while Mohammad Shahzad smashed 22 runs off 16 and opener Rohan Mustafa contributed 20.
Porterfield and Stirling found the boundary regularly as they hit 61 off just 42 balls, before Stirling fell to Amjad Javed.
Poynter's dismissal for 10, with the score at 91, signalled the beginning of Ireland's demise as wickets fell at regular intervals thereafter.
For skipper Porterfield, it was his third half-century in T20 internationals, and his 72 off 60 balls included eight fours and a six.
However with 13 needed off 11 balls, he was run out by wicketkeeper Swapnil Patil.
Nine were needed off the last over, but Max Sorensen was run out off the first ball, and George Dockrell bowled from the second.
That left Rankin and Tim Murtagh at the crease, and the pair managed just a scrambled bye and a single as the hosts completed a superb comeback win.
Mohammad Naveed, Ahmed Raza and Mohammad Shahzad collected two wickets apiece.
The match was part of Ireland's preparation for their World Twenty20 campaign, which starts against Oman in the Indian city of Dharamsala on 9 March.
The squad will now have a week's break before travelling to Chandigarh for a training camp before a warm-up game against Hong Kong on 3 March.
A dramatic batting collapse contributed to Ireland's five-run loss to the United Arab Emirates in their second Twenty20 international.