The Mountain Hawks (7-0, 1-0) have won 15 straight regular-season games dating back to last season, good for the longest active streak in the Football Championship Subdivision.
Barket, who finished with 123 yards rushing, tied the game 7-7 with his 1-yard touchdown run.
Georgetown (3-4, 0-1) regained the lead, 14-7, behind Jeremy Moore's 80-yard interception return for a touchdown with 3:52 left in the first half. The Hoyas' defense forced turnovers (four interceptions, one fumble) on five of Lehigh's first eight possessions.
The Mountain Hawks' offense rallied in the second half, opening the third quarter with a 9-play, 76-yard scoring drive. Keith Sherman capped the drive with a 1-yard touchdown.
Peery's field goal with 10:58 remaining sealed the win.