Edwards, who was coming off a 1-for-5 for 18 yards performance in the Bills' preseason opener against Washington, tossed both of his touchdown passes to tight end Robert Royal.
And rookie Leodis McKelvin, Buffalo's No. 1 pick, returned a kickoff 95 yards untouched for a touchdown as the Bills hung on for the win.
"We have to get our regular season legs under us pretty quickly and become a better tackling team," said head coach Mike Tomlin. "Nothing is broken schematically. Guys are where they are supposed to be. We don't have enough guys making sure tackles."
The Bills also rushed for 159 yards on 31 carries, including 25 on five attempts for starter Marshawn Lynch against the first-team defense in the first quarter.
Backup Byron Leftwich, signed on Sunday, completed 5-of-11 passes for 41 yards, while third-stringer Dennis Dixon was 6-of-15 for 79 yards and also had an electrifying 47-yard touchdown run in the fourth quarter that nearly brought the Steelers back.
Dixon's Hail Mary pass to fellow rookie Micah Rucker was knocked from Rucker's hands in the end zone to end the game.
Rookie Rashard Mendenhall scored the Steelers' other touchdown, diving into the end zone from six yards out in the third quarter. Mendenhall, the Steelers' top draft pick, had 30 yards on 11 carries.