Rahmat was backed up by Hashmatullah Shahidi, whose 82 from 100 balls, saw the duo put on 184 for the third wicket.
Afghanistan skipper Asghar Afghan pitched in with an unbeaten 21 to help his side to victory.
Ireland had won the toss and chose to bat, Paul Stirling hitting 128 from 132 balls to help his team get to 259/9 from their 50 overs.
Naveen-ul-Haq claimed Stirling's wicket, ending with 4-42 off his 10 overs.
