He hadn't allowed a hit until Goodrum, who entered the game with 14 career hits, singled up the middle to lead off the third inning.
The Tigers added another run in the seventh when Mikie Mahtook doubled home Iglesias, who had reached base after getting hit by a pitch.
Shane Greene pitched a 1-2-3 top of the ninth inning to earn the win.
The Seattle Mariners are looking to end baseball's longest postseason drought, but their hopes for contention just took a big blow.
Cano will visit a hand specialist Tuesday in Philadelphia and expects to require surgery to fix the broken bone at the base of his right pinkie finger, Greg Johns of MLB.com reports.
The Mariners said Cano was diagnosed with a fractured fifth metacarpal.
Andrew Romine took his place at second base, defensively, but the Mariners will make a corresponding move to fill Cano's place on the roster with a trip to the disabled list in store.
Marco Gonzales (3-3) allowed four runs and 10 hits in six innings. Over his last eight appearances, he's allowed just one walk and one hit (a single) in 8 innings, striking out seven.
Kyle Seager scored the Mariners' first run on a solo home run - his seventh of the season. On the mound, they have been rather poor as they come in ranked 23rd in the league in ERA at 4.66, while also ranking 16th in WHIP at 1.32 and 13th in K's with 335.