Murtagh followed up his four wicket haul with an unbeaten 19, sharing an unbroken 9th wicket stand of 29 with Andy McBrine (14*).
Murtagh's best bowling figures in an ODI saw Zimbabwe collapse from 159 for 4 to 187 all out - losing their last six wickets for just 28 runs.
Craig Young (1-35) - replacing John Mooney in the starting line-up - got the Irish off to a dream start by claiming a wicket in the first over after skipper William Porterfield won the toss.
Sean Williams stroked 8 boundaries in a fluent 51 before being dismissed by Paul Stirling (1-17), and at 88 for 4 the contest was finally balanced.
Zimbabwean skipper Elton Chigumburu and the ever consistent Sikandar Raza added 71 for the fifth wicket, before Andrew Balbirnie held a fine catch on the boundary to account for Chigumburu (34).
The dangerous Raza fell immediately after reaching his half century (56 balls, 4 fours, 1 six) to an injudicious stroke - Murtagh again the successful bowler.
George Dockrell then secured two brilliant run outs in the space of six balls as the Zimbabwean innings imploded.
Kevin O'Brien (1-14) wrapped up the Zimbabwean innings on what looked a well below par 187 at the half-way stage.
William Porterfield got the Irish run chase off to a blistering start as 18 runs came off the first over, but fell soon after for a run-a-ball 20 (3 fours).
Ed Joyce (7) went cheaply, but the Middlesex pairing of Paul Stirling and Andrew Balbirnie looked to be steering Ireland towards a comfortable win as they added 48 to take Ireland to 94 for 2 approaching the half-way stage.
However Balbirnie (24) ran himself out for the second game in succession, and when Niall O'Brien fell cheaply soon after the contest was once again finely balanced.
A restrained Stirling reached his 50 from 92 balls (4 fours), and when Gary Wilson (19), Kevin O'Brien (10) and George Dockrell (13) were all dismissed despite getting starts, Zimbabwe had edged ahead with Ireland on 160 for 8.
Step forward the 'Lambeth Lara' aka Tim Murtagh, who settled the Irish nerves with two boundaries in his 19 not out, helping secure the vital win in conjunction with Donemana's Andy McBrine who played another crucial cameo.
The consolation win keeps Ireland in 11th place in the ICC rankings - just ahead of Afghanistan who start their series against Zimbabwe later this week.
The Irish are next in action on Saturday when they start their four-day match against Zimbabwe A. That game is being used as preparation for the InterContintental Cup game against Namibia which gets under way in Windhoek on Saturday October 24th.
ZIMBABWE v IRELAND, 3rd ODI,
At Harare Sports Club,
ZIMBABWE 187 all out (49.2 overs, S Williams 51, S Raza 50, E Chigumburu 34, T Murtagh 4-32)
IRELAND 189/8 (46.5 overs, P Stirling 50, A Balbirnie 24, W Porterfield 20, W Mazakadza 2-31, T Mutombodzi 2-33, T Muzabarani 2-43)
Ireland beat Zimbabwe by two wickets