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Chasing a revised target of 76 in 10 overs after rain, Scotland won by eight wickets with 12 balls to spare thanks to Matthew Cross' rapid 22.
Mark Chapman made 40 in Hong Kong's modest 127-7 as Matt Machan took 2-26.
Earlier on the same ground, Afghanistan beat Zimbabwe to top the group and qualify for the Super 10s.
Scotland's win earned them third place in the group, with Hong Kong bottom after losing all three matches.
Beaten by Afghanistan and Zimbabwe in their opening two games, Scotland's sequence without a win at a major tournament stretched back to the 1999 World Cup.
"It's a great day for Scottish cricket," said man-of-the-match Machan, who hit the winning runs to finish 15 not out off four balls.
"We haven't played our best cricket this week but we've got the monkey off our back and we're very happy."
Scotland captain Preston Mommsen added: "It's huge - it's been a long time coming.
"The boys are absolutely chuffed to get on the board. There will be a few cold ones cracked open tonight."
Scotland's victory owed much to the discipline of their bowling attack, with slow left-armer Con de Lange and seamer Richie Berrington conceding fewer than five runs an over.
Although Chapman helped Hong Kong recover from 33-3 alongside Anshuman Rath, they struggled to find the boundary until Nizakat Khan hit two sixes in his 17 off 10 balls late on.
Scotland's chase, delayed by a second rain delay lasting 45 minutes, was little more than a formality.
Cross provided the early impetus, opener Kyle Coetzer made a patient unbeaten 20 and Machan wrapped up victory with a six over deep mid-wicket.
Scotland recorded their first win in 21 attempts at a major global tournament by beating Hong Kong in their final World Twenty20 group match in Nagpur.