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).
A months-old Facebook post has cost the Nebraska football stadium announcer his job one day after his hiring was announced.
Former running back Lawrence Phillips has been charged with first-degree murder in the April death of his cellmate at Kern Valley State Prison.
Michigan coach Jim Harbaugh said Tuesday he has informed competing quarterbacks Jake Rudock and Shane Morris of his decision on a starter for the Wolverines' opener Thursday night at Utah.
Michigan State quarterback and Heisman Trophy hopeful Connor Cook said he was disappointed his teammates didn't vote him among the team's three captains this season.
Former Oklahoma State return man and running back Ty Hill, who was dismissed from OSU's program after his arrest on domestic violence charges, has enrolled at West Alabama.
UCLA junior cornerback Ishmael Adams has been suspended indefinitely following a robbery arrest early Sunday morning, coach Jim Mora said Tuesday.