Fanning, who made 9 of 11 field goal attempts, scored seven points in the extra period.
Jarred Jones made a layup with 3:08 left in OT to give Loyola (4-13, 3-3) a three-point lead.
A bucket by Nick Havener, a 3-point play by Fanning and two free throws by Mosely put the Terriers up 83-79.
Jones made a pair of free throws to pull the Greyhounds within two with 58 seconds to go but Fanning made 4 of 4 foul shots from there and, after Loyola's Nevell Provo missed a potential tying 3, grabbed a rebound to seal it.
Jones led Loyola with 20 points and 11 rebounds.