Montgomery found the endzone twice in the first half and again in the fourth quarter to clinch the visitors’ third win of the season.
The Packers were able to impact the scoreboard early, gaining a 3-0 lead with a field goal from their opening drive.
However, they had to wait a long time between drinks from there, as the Lions hit back with three touchdowns and two field goals to take a 27-3 lead into half-time.
Green Bay were able to stem the tide in the third quarter, holding Detroit to nil while picking up a touchdown through wide receiver Christian Watson.
They inched closer at the start of the fourth thanks to a nine-yard touchdown run from quarterback Jordan Love, but their comeback bid was officially ended when Montgomery picked up his third touchdown with six minutes remaining.
Love finished with 246 passing yards to go along with a touchdown and two interceptions, while Detroit quarterback Jared Goff threw for 210 yards, a touchdown and an interception of his own.