The Blackbirds (20-7, 14-1) led 79-73 with just 35 seconds left to play after Jason Brickman converted one of two free throws.
The Blackbirds dominated overtime, scoring the first 10 points and converting 12 of 15 free throws during the period.
C.J. Garner, who finished with 17, made six free throws in overtime and his jumper with 36 seconds left gave LIU a 103-89 lead.
Gibson led Sacred Heart (13-15, 7-8) with 24 and Justin Swidowski chipped in with 23.