Eddington ran for 129 yards on 11 carries and Hall added 117 on 24 attempts.
Gino Guidugli completed only 13-of-25 passes for 121 yards, but threw his 39th career touchdown pass to tie the school mark set by Danny McCoin from 1984-87.
Tedric Harwell rushed for two touchdowns for the Bearcats (1-0, 1-0 C-USA), who defeated East Carolina, 42-26, in last year's regular season finale to gain a share of the Conference USA title.
Cincinnati opened a 14-0 first-quarter lead on one-yard TD runs by Hall and Harwell.
Guidugli tied McCoin's record with a 29-yard TD strike to Dennis Hart with 23 seconds left in the second quarter that gave the Bearcats a 21-0 halftime lead.
East Carolina, playing its first game under coach John Thompson, got its only points on Cameron Broadwell's 36-yard field goal with 11:36 left in the third quarter.
The Pirates were held to 203 total yards, including just 57 on the ground.
Eddington's 23-yard TD run and Chet Ervin's 42-yard field goal gave the Bearcats a 31-3 lead heading into the final period.
The Bearcats recorded a safety in the fourth quarter before Harwell's 10-yard TD run with 1:25 remaining capped the scoring.
Corey Robinson is returning for his senior year on the gridiron at Notre Dame while also assuming the responsibilities of student-body president.
Baylor will hire more counselors and provide additional training for faculty in an effort to improve the school's response to sexual assault incidents, school regents said Friday.
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.