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.
Answering questions about the hype around two young quarterbacks, whether Stanford will sooner or later fall back into mediocrity and more.
FOX Sports' Clay Travis joins The Paul Finebaum Show to discuss the possibility of losing Tigers football to budget cuts.
The Naismith Memorial Basketball Hall of Fame announced 14 finalists for the Class of 2016 on Friday, including Allen Iverson, Tom Izzo, Shaquille O'Neal and Yao Ming.
Discussing the transfer of Barry J. Sanders to Oklahoma State, what the latest staff movement means for Charlie Strong and TCU's linebacking corps.
Stephanie Grace of the Baton Rouge Advocate calls into The Paul Finebaum Show to talk about the impact of Louisiana's budget crisis on LSU.
A list of when every Big 12 team will start spring ball, and the date of every Big 12 spring game.