Last week's opening game in the five-match contest was washed out, and Afghanistan won Friday's second ODI in Bready by 92 runs.
Ireland scored 206 in Monday's third match but three wickets apiece from Rashid Khan (3-44) and AM Ghazanfar (3-29) kept them in check after they were put in to bat.
The home team were in deep trouble at 128-8 before Gavin Hoey (36) and Jai Moondra (31) put on 76 for the ninth wicket, but they were all out with 3.3 overs remaining.
Afghanistan started their reply steadily and had reached 53 before opener Ibrahim Zadran fell for 12.
Sediqullah Atal (27) and Hashmatullah Shahidi (22) chipped in with useful runs as Rahmanullah led the way, smashing five sixes and two fours before he was caught by Mark Adair off Hoey.
Ireland had a glimmer of hope when Adair removed Azmatullah Omarzai to leave Afghanistan 176-7, but Rashid's 37 not out in an impressive all-round display helped the visitors home.
Afghanistan's victory also secures their spot in the 2027 ODI World Cup in Southern Africa on account of their now unassailable world ranking (eighth), condemning the West Indies to next year's global Qualifier.
The two sides meet again in Belfast on Wednesday.