The Hornets (6-4, 5-2 Mid-Eastern Athletic Conference), who have won five of their last six games, led 21-19 early in the fourth quarter.
Elko completed an 83-yard scoring pass to Tarpley to extend the lead, then Dae-Hon Cheung's 6-yard score made it 35-19 with 4:01 remaining.
On the ensuing kickoff, Antwon Chisholm scampered 59 yards, and Travis Champion threw his third TD pass of the day, a 30-yarder to Rashad Riddick, on the next play. After Hampton made the two-point conversion, Delaware State ran out the clock.
Tarpley, whose other TD was a 48-yarder, had five receptions for 171 yards and Nagee Jackson added scoring runs of 1 and 2 yards.
Jeremiah Schwartz rushed 136 yards for the Pirates (2-7, 2-5).
Ohio State quarterback Cardale Jones was taken to a hospital Wednesday night and treated for a severe headache, according to multiple media reports.
The Florida Gators will start Treon Harris at quarterback in Saturday's opener vs. New Mexico State over redshirt freshman Will Grier, coach Jim McElwain said Tuesday.
Baylor University has hired a Philadelphia law firm to independently investigate the university's handling of sexual assault complaints, including one against defensive end Sam Ukwuachu.
Oregon State and former coach Mike Riley face a federal Title IX lawsuit filed by a woman who said she was raped as a result of the coach's failure to correct a sexually violent culture.
James Madison players will wear decals on their helmets during Saturday's season-opening game in memory of the two journalists who were killed during a live television broadcast.
Los Angeles prosecutors have charged a man in connection with the theft of the University of Southern California's copy of O.J. Simpson's Heisman Trophy more than 20 years ago.