WATCH: Richard Cockerill enjoys a first win in Belfast - with Edinburgh
The win means they overtake their hosts in third place in the table and it was a victory head coach Richard Cockerill enjoyed - his first ever in Belfast.
The Scots now have an edge on the Irish province in the race for the top three play-offs, though both sides are on equal points in the table after Ulster, who slipped to fourth, gained a losing bonus point.
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Hide AdTries from winger Jason Harries and lock Lewis Carmichael along with two conversions from Sam Hidalgo-Clyne, along with Weir's drop goal, earned the Scots the four points.
All Ulster's points came from John Cooney.