Early on in the First Round Group B match at the Al Amerat Stadium in Muscat things were looking bleak for the Scots against their sixth in the world opponents as they slipped to 53-6 with the bat.
However, 31-year-old South African-born Greaves - who qualifies for Scotland via residency - took things by the scruff of the next in just his second ever T20 international appearance.
He scored 45 runs from 28 runs to haul the team up to 140-9 from their 20 overs and then took two key wickets with his leg spin in the second half of the match.
The other bowlers - and the fielders with some smart catches - also stood up to the mark as the Scots sensationally won with Bangladesh ending on 134-7.
If the Scots can back up this showing against Papua New Guinea on Tuesday and Oman on Thursday then they will finish in the top two in the group and progress to the ‘Super 12’ phase where they would face the best teams in the world.
“It feels incredible to be a Scottish cricketer right now,” Greaves, who only made his international debut in this format on October 8, said.
“I am really enjoying being a part of this team and the spirit in the group is great.
“When I came in with the bat myself and Mark Watt had to just try and rebuild things and see what sort of total we could get the team up to.
“It feels unbelievable that I could be one of the people to help us get to a competitive total.
“We gave ourselves something to bowl at with 140 on the board and we then backed our bowling line-up to do the job and they did.”
Scotland lost skipper Kyle Coetzer, Matthew Cross and George Munsey, the latter for 29, to leave them on 46-3.
Richie Berrington and Michael Leask then both fell to Shakib Al Hasan - who is now the world’s leading wicket taker ever in this format - and when Calum MacLeod was bowled it was 53-6.
Greaves and Watt showed no nerves though and played with freedom, hitting some much-needed boundaries and running well to take the side up and over 100 and their 50 partnership came in the 17th over.
Watt went for 22 off 17 balls, but Greaves carried along on his merry way.
He was out in the final over and his runs had helped the team up to a competitive total.
Scotland needed wickets early on in the Bangladesh reply and Josh Davey and Brad Wheal reduced them to 18-2.
Al Hasan and Mushfiqur Rahim then took their side up to 65, but the latter fell to Greaves with his first ball in the 12th over thanks to a brilliant catch from MacLeod.
Soon after Greaves bowled Rahim and with four overs remaining Bangladesh were on 92-4 and still needed 49 runs to win.
Watt got his 50th T20 international wicket in the 18th over and Bangladesh needed 32 off 12 balls.
Bowlers Wheal and Safyaan Sharif then kept their cool in the last two overs to complete a famous triumph.