The African nation restricted the Scots to 132-6 with neat bowling led by Tendai Chatara (2-14) and Richard Ngarava (2-28) before cracking through the run chase with nine balls to spare.
Zimbabwe have been part of five previous World Cup tournaments, but never went beyond the first round.
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But they have improved since coach Dave Houghton, Zimbabwe’s first Test captain in 1992, joined the team for his second stint in July.
They beat Ireland to open their latest World Cup campaign and then lost to the West Indies before booking their place against the Scots.
Scotland captain Richie Berrington won the toss and elected to bat, but they were immediately on the back foot, losing Michael Jones, who had hit a masterful 86 against Ireland, in the first over.
Sikandar Raza took an excellent over-the-shoulder catch off Chatara’s bowling to dismiss him.
Another spectacular diving catch was held by Wessly Madhevere at mid-wicket to remove Matthew Cross for one four overs later, leaving Scotland 24-2 in the fifth over.
Opener George Munsey, on the other hand, remained calm and built a 40-run partnership with Berrington before the captain removed Raza for 13 against Milton Shumba in the deep.
The run rate slowed in the face of economical bowling as Munsey reached a ninth T20 half-century with a single, only to fall five balls later for 54, caught by Shumba of Ngarava, before Calum MacLeod (25) and Michael Leask (12) ) added important late runs.
Zimbabwe started the chase with a four but then lost two wickets in eight balls.
After hitting the boundary, Regis Chakabva was caught lbw by Brad Wheal and then Madhevere departed without scoring, dragging a Josh Davey delivery onto his stumps.
Sean Williams scored 35 in a partnership with Ervine, who missed their last game with a “slight asthmatic attack”, but the stand was broken when Michael Leask came into the attack and he was caught by Wheal for seven.
This left Zimbabwe at 55-3 after 10 overs.
But Ervine kept his focus to reach a ninth T20 half-century, happy to play second fiddle to the dangerous Raza (40 off 23) who swung the bat before being caught behind by Cross van Davey.
With Ervine out, they needed 14 from the last three overs, with Chris Greaves and Davey achieving the feat.