The Blazers (4-7, 3-4) jumped out to a 17-3 lead with 10:24 remaining in the second quarter thanks to a 52-yard touchdown pass from Ellis to Jeffrey Anderson, a 1-yard scoring run by Shed and a 25-yard field goal by Josh Zahn.
Ellis connected with Shed on a 54-yard touchdown pass 1:53 before halftime to extend UAB's lead to 24-6.
Paulo Henriques kicked field goals of 44, 43 and 36 yards for Memphis. The 36-yarder, which came with 20 seconds left in the second quarter, pulled the Tigers (1-10, 0-7) within 15 points, 24-9, at halftime.
Ellis hit Patrick Hearn for a 72-yard touchdown pass in the fourth.
Memphis has now lost 10 games in consecutive seasons and three times since 2006.