The Demons got the better of the Swans early, breaking out to a 10-point lead at the first change at Deakin Reserve.
A four-goal-to-one second term gave the Swans a nine-point lead of their own at half-time before each team kicked four goals in the third quarter as the visitors took a one-goal buffer into the final change.
Despite United winning the last quarter, the Demons couldn't get over the line as the Swans held on to win 11.10 (76) to 11.7 (73).
Zac Alderton kicked three goals for the Swans, while Zac Clohesy and Steele Simpson each added two.
United's Elliot Cavallaro and Edward Adams kicked three apiece to give the Demons a sniff.
Swans coach Paul Hawke said the victory was reward for effort from his group.
“It's been building for a few weeks,” Hawke said.
“A couple of games ago we were inaccurate and against Kyabram we were in it until the last 10 to 15 minutes.
“We had been playing some good footy, but hadn't come away with the ultimate reward of winning the game.”
Hawke gave credit to the Demons, who never gave in.
“Even though it seemed that we controlled the game for long periods, they would score and score quickly to keep themselves in the game,” Hawke said.
“It was a great win for the club, for our supporters and our sponsors and I'm hoping we have a few more wins in us for the rest of the season.”
While he admitted it was an even performance that got them over the line, Hawke did single out a couple of players who led from the front.
“(Andrew) Riordan and BJ Squire were our two best players, but right across the board we had contributors across the day,” he said.
That's what we are looking for — that even performance — and everyone did their part yesterday (Saturday).”
The Swans are in 11th spot on the table and join Shepparton and Tatura on two wins and eight losses.
United, which has a 4-6 record, has dropped to ninth, but is half-a-game from fifth.
The Swans next play Mooroopna at home, while United is up against the Bears.
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