All three of Oklahoma's losses came in a three-year span when the Sooners went 33-0 against all opponents except the Hurricanes.
Bradford completed 19-of-25 passes for 205 yards in his second career start, and Malcolm Kelly caught three of his touchdown passes as the Sooners (2-0) evened the all-time series at three games each.
Oklahoma built a 21-3 first-half lead on Bradford's touchdown passes of 23 and 24 yards to Kelly and Smith's 61-yard return of a Javarris James fumble -- the third-longest fumble return in Sooners' history.
Neither starter Kirby Freeman nor backup Kyle Wright could mount enough of a comeback, and Miami (1-1) suffered its most lopsided defeat since a 66-13 loss to Syracuse on Nov. 28, 1998.
Under Freeman's guidance, Miami settled for a field goal after getting a first-and-goal at the 8-yard line following Oklahoma sophomore Derek Shaw's snap over punter Michael Cohen's head. Through seven offensive drives, Freeman completed only two passes for 10 yards and Miami had only one first down.
That led first-year coach Randy Shannon to make a change.
Wright, who split starts with Freeman last season, entered and led an 18-play, 52-yard scoring drive that was extended by a fake field goal and two pass interference penalties in the end zone. Wright finished it with a 6-yard touchdown pass to Ryan Hill to pull Miami within 21-10 before the half.
The Hurricanes had a chance to cut into the deficit after Chris Brown fumbled on Oklahoma's opening drive of the second half, but again settled for a Francesco Zampogna field goal.
Sparked by Juaquin Iglesias' 43-yard kickoff return, Oklahoma responded immediately on a shortened field and tight end Jermaine Gresham caught Bradford's third touchdown pass for a 7-yard score.
Bradford and Kelly hooked up again for a 30-yard scoring strike, with Kelly making an over-the-shoulder reception, to make it 38-13 early in the fourth quarter. Kelly finished with 102 yards receiving on four catches.
Bradford then added a 9-yard TD pass to fullback Dane Zaslaw to match the school record set by 2003 Heisman winner Jason White and 2000 Heisman runner-up Josh Heupel.
Wright was 7-for-14 for 65 yards. Freeman, who returned late in the third quarter, was 3-for-9 for 17 yards.
James and Graig Cooper, who combined for 215 yards rushing in a 31-3 win against Marshall last week, were held to 54 yards.