Harnish was 20 of 25 for 211 yards and helped the Huskies (4-2, 2-0 Mid-American Conference) rally from a second-quarter deficit with 21 unanswered points.
The Owls (4-2, 1-1) led 17-10 after Chester Stewart threw a 31-yard touchdown pass to Evan Rodriguez late in the first half.
Northern Illinois then tied the score on a 10-yard TD strike from Harnish to Landon Cox with 30 seconds left before halftime. The Huskies took the lead for good on Harnish's 4-yard scoring pass to Nathan Palmer with 4:15 remaining in the third, and Spann, who got off to a rough start, added a 6-yard TD run in the fourth.
Spann fumbled on the first play from scrimmage, and Temple quickly capitalized as Matt Brown scored on a 25-yard run 30 seconds into the game.
Northern Illinois replied with an 80-yard drive, capped by a 9-yard touchdown run by freshman Akeem Daniels on his first career carry.
After the teams traded field goals, Stewart connected with Rodriguez to give Temple the lead with 3:59 left in the half.
After Owls kicker Brandon McManus missed a field goal early in the third, the Huskies marched 75-yards to claim their first lead.
Stewart was 13 of 27 for 148 yards and Brown led Temple with 93 rushing yards.
Harnish finished with 79 rushing yards for the Huskies.