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).