Beavan (9-8) walked two and struck out one while giving up five hits in seven innings.
The Twins amassed three of those hits in the first inning for a pair of runs and a 2-1 lead.
That was evaporated, however, by Seattle's four-run sixth inning that beat Brian Duensing (3-10).
The Mariners' rally included two singles, two walks and back-to-back sacrifice flies by Kyle Seager and Jesus Montero. Trayvon Robinson capped the rally with a two-run base hit that make the score 5-2.
Duensing ended up surrendering four runs on four hits in 5 1/3 innings. He also walked two and struck out three.
The Twins got within 5-4 when John Willingham hit his 33rd home run of the season, a two-run shot in the eighth inning.
Seattle closer Tom Wilhelmsen in the ninth walked two and battled back from a 3-0 count to entice a ground out from Ben Revere to end the game for his 21st save.
The Mariners got an RBI single from Seager in the first but Minnesota answered with its first-inning rally as Alexi Casilla drove in the first run and scored the second on a Justin Morneau sacrifice fly.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys