Chasing a modest target of 152 from their allocation of 50 overs, West Indies lost five wickets before they achieved victory in the 39th over.
New West Indies captain Brian Lara had the privilege of hitting the winning runs, when he lofted a delivery from Ryan Higgins over wide long-on for a six to give his side a 1-0 lead in the seven-match series.
Lara's unbeaten 40 from 43 balls that contained three fours and one six was the top score for the home team.
Prosper Utseya was the most successful Zimbabwe bowler with three wickets for 35 runs from his allotment of 10 overs.
Sent in to bat on a helpful pitch, Zimbabwe had been restricted to 151 for nine from their allocation of 50 overs.
Justice Chibhabha hit five fours in the top score of 55 from 99 balls and featured in a vital stand of 64 for the second wicket with Zimbabwe captain Terrence Duffin, who scored 26.
Four West Indies bowlers - Jerome Taylor, Dwayne Bravo, Dwayne Smith and Chris Gayle - collected two wickets apiece.
West Indies suffered an early casualty, when Runako Morton, opening the batting, was caught at second slip for a duck off the last ball of the first over bowled by Edward Rainsford.
Ramnaresh Sarwan came to the wicket and announced himself with a crisp extra cover drive off Rainsford for a boundary.
Rainsford was left disappointed however, when wicketkeeper Brendon Taylor muffed a chance when Sarwan was on 12, the same score on which he offered a chance to Rainsford at third man off Mahwire, but the fielder could not get under the skier.
Sarwan's luck eventually ran out, when on 14 he failed to beat Prosper's direct hit from square leg in the 12th over.
Opening batsman Chris Gayle was caught at short third man for 35 off Prosper, and fellow left-hander Shivnarine Chanderpaul was caught behind top-edging a cut at a delivery from Prosper to leave West Indies 77 for four.
Ramdin and Lara added a crucial 44 for the fifth wicket to put West Indies back on track.
Ramdin had been fortunate that his leading edge off a delivery from Chibhabha in the 26th over flew over the bowler's head and not into his hand, but Utseya bowled him for 37 in the 33rd over to leave Lara and Dwayne Bravo to guide the team to victory.
Earlier, the visitors looked set to make a competitive total when Chibhabha and Duffin added 64 for the second wicket.
Once they were removed, there was neither substance nor stability from the rest of the batting, as the Zimbabweans lost their last six wickets for 30 runs in the final 10 overs.
The visitors suffered an early setback, when opener Piet Rinke was caught at square leg for one in the second over.
Duffin had been fortunate in the fourth over, when on six he gloved a rising ball from Taylor and wicketkeeper Denesh Ramdin spilled a straightforward chance.
Ramdin however, made no mistake the second time around just when it looked like Duffin and Chibhabha were getting ready to play their strokes.
Duffin was caught behind for 26 off Dwayne Smith edging a backfoot drive off the last ball of the 27th over.
Chibhabha continued to plod away and reached 50 when he sliced Taylor through backward point for two, but he became reckless after arriving at the milestone and was caught at long-off off Smith in the 37th over.
The series continues on Sunday when the two teams contest the second match at the same venue before heading to Georgetown for back-to-back matches on May 6 and 7.
They will play the first limited-overs international under lights in the Caribbean at Gros Islet's Beausejour Stadium on May 10 before the series ends in Port of Spain on May 13 and 14.