With the score tied 17-17, Aaron Rodgers led the Packers on a seven-play, 59-yard drive, capped by Harris' 14-yard touchdown run with 10:48 left in the contest.
Mason Crosby added the second of his two field goals with 4 minutes left and Green Bay (9-4) recorded its second win in two weeks, pulling one game ahead of the Chicago Bears (8-5) atop the NFC North.
Rogers completed 14-of-24 passes for 173 yards on a frigid, snowy night at Lambeau Field, while also running for a 27-yard touchdown in the third quarter.
Randall Cobb hauled down seven passes for 102 yards for the Packers.
Matthew Stafford was 27-of-45 in passing for 264 yards, a touchdown and an interception for Detroit (4-9), which fell to 0-5 against NFC North opponents.