Liberty, an FCS school, led 41-31 early in the fourth quarter, after Calvert scrambled into the end zone from 8 yards out.
Baylor responded with a quick, four-play 75-yard drive that was capped by John Lovett's 3-yard touchdown run that cut the Liberty lead to 41-38 with 10:39 to play.
But Calvert answered again, hooking up with Spencer Jones on a 26-yard touchdown pass with six minutes left, and the Flames held on for the biggest early shocker of the young college football season.
Baylor senior quarterback Anu Solomon completed 14 of 29 passes for 278 yards with three touchdowns, but couldn't connect on a last-play Hail Mary that was picked off by Liberty's Zach Smith.
The two teams combined for 1,117 yards of total offense.
Calvert finished 44 of 60 for 447 yards, three touchdowns and no interceptions.
Liberty receiver Antonio Gandy-Golden had 13 receptions for 192 yards and two touchdowns.
Baylor receiver Chris Platt caught three passes for 102 yards and two touchdowns, including a 75-yard hook up with Solomon that gave the Bears a 31-27 lead with five minutes left in the third quarter.
Calvert hit Gandy-Golden on a 13-yard touchdown pass on the Flames' ensuing drive, and Liberty took a 34-31 lead into the fourth quarter.
The game was tied 17-17 at halftime. Solomon threw a 45-yard touchdown pass to Denzel Mims, and JaMycal Hasty had a 14-yard touchdown run for the Bears. But Baylor still needed a 39-yard field goal from Connor Martin late in the second quarter to pull even with the Flames at halftime.
Baylor went 7-6 last year, capped by a bowl win over Boise State. But the Bears' season was marred by an off-the-field scandal that was centered on the mishandling of multiple sexual assault complaints and led to the firing of coach Art Briles.
Rhule accepted the Baylor position after a successful, four-year tenure at Temple.