The hosts restricted Ireland to 219-8 from their 50 overs, Gary Wilson scoring an unbeaten 70 and Ed Joyce adding 53.
Ireland reduced Zimbabwe to 171-7 before Raza's unbeaten knock helped them to victory with an over to spare.
The second game takes place on Sunday before a potential decider on Tuesday.
Paul Stirling (24) and Kevin O'Brien (16) contributed to a modest Ireland total while John Nyumba (2-35) and Wellington Masakadza (2-45) each picked up two wickets.
Spinner George Dockrell was the pick of the Irish bowlers with 2-29.
Ireland hope to end the series still ahead of Zimbabwe in the world rankings.
The Irish started the series in 10th and Zimbabwe 11th, with the tourists holding a two-point advantage over their rivals.
Zimbabwe fell to a 2-1 series defeat by Pakistan in Harare last week.

Victims reacted angrily to the judge's decision in the Pretoria High Court to extend bail pending an appeal hearing on 19 June.
The former Grand Slam champion will now be placed under house arrest.
Hewitt, 75, was found guilty of raping and sexually assaulting girls he coached in the 1980s and 1990s.
Judge Bert Bam criticised the Australian-born player at his sentencing on Monday for not showing remorse during his trial.
He said justice must be seen to be done, even for ageing offenders.
One of the victims who gave evidence at the trial said that Hewitt had assaulted her when she was just 12 years old.
The judge said at the time of Hewitt's conviction that the striking similarities between the three victims' testimonies showed that his conduct was calculated.
Speaking at the sentencing hearing in Pretoria, Hewitt's wife Delaille had begged the court to show mercy and not award a custodial sentence.
Hewitt had testified that he suffered from poor health.
The BBC's Nomsa Maseko, in Johannesburg, says the tennis champion has also been ordered to pay a fine to the South African Department of Justice, which will help fund its campaigns against sexual abuse.
Hewitt's lawyers say they will appeal against his conviction and sentence.
He was suspended from the International Tennis Hall of Fame in 2012 over the allegations of sexual abuse.
Hewitt played initially for Australia, later moving to South Africa and taking citizenship there.
He was a member of the South African team which won the Davis Cup by default in 1974.
India, their opponents, refused to travel to South Africa for the final as a protest over the government's apartheid policies.

Connors, 21, began his career with Fulham and joined the Daggers in 2011 on a two-year scholarship.
"He is versatile, can play in two or three positions and he knows Scott Doe and Luke Howell from their time at Dagenham," said boss Luke Garrard.
"Jack's signing was important for us in terms of quality and squad depth."
Craig Ervine and Sikandar Raza both hit 60 as Zimbabwe clinched a two-wicket win over Ireland in the first of three one-day internationals in Harare.
[NEXT_CONCEPT]
Former tennis star Bob Hewitt has been granted bail by a South African court, after receiving a six-year prison sentence for raping underage girls.
[NEXT_CONCEPT]
Boreham Wood have signed defender Jack Connors on loan from League Two side Dagenham & Redbridge until the end of the season.