Franklin's three-run shot came after Michael Saunders had connected for a bases-empty homer earlier in the inning and handed the Mariners a 5-2 lead.
The rookie second baseman later added his own solo shot, going 3-for-4 overall as the Mariners earned a four-game split with the Twins.
Erasmo Ramirez (2-0) got the decision despite yielding four runs on seven hits over six innings.
Danny Farquhar supplied two scoreless innings of relief and Tom Wilhelmsen nailed down Seattle's win with his 24th save.
Minnesota's Kyle Gibson (2-3) was roughed up for five runs on seven hits in five innings.
Chris Colabello went 2-for-4 with a two-run homer for the Twins, and narrowly missed recording a three-run shot in the third when Seattle's Endy Chavez leaped above the right field wall for a long out.