Signed during the offseason to bring stability to Cleveland's troublesome position and mentor Manziel, McCown connected with Travis Benjamin on a 2-yard TD in the first quarter. After scrambling from pressure, McCown was about to throw the ball away when he saw Benjamin waving and wide open.
"I thought, 'Is he back there by his self?'" McCown said. "He looked lonely, so I threw him the football."
Griffin III went 4 of 8 passing for 36 yards during two possessions with the starters. Griffin would have thrown a TD pass, but wide receiver Pierre Garcon dropped a long toss.
The win was costly for Washington, which lost starting tight end Niles Paul to a broken ankle and running back Silas Reed sustained a knee injury.
"I'm sick about it," coach Jay Gruden said of Paul's season-ending injury. "He means a lot to this team."
McCown, who went just 1-10 as a starter last year with Tampa Bay, was sharp during his one series. The 36-year-old completed all five passes and displayed a veteran's poise on his TD pass to Benjamin.
Washington's first-team offense didn't produce a touchdown in the preseason last year, a drought that foreshadowed Griffin's struggles in his third season.
"You're going to get hit every now and then," Griffin said. "I thought we did a good job moving the ball. Get in, get out and that's all you can ask for."
Manziel has shown maturity on and off the field following a disastrous rookie season, and the 2012 Heisman Trophy winner made a smart play when he tucked the ball away and darted 12 yards for his TD. But unlike a year ago, Manziel, who was limited earlier this week by a sore elbow, didn't celebrate by flashing his signature "money" sign, a gesture he has promised never to do again.
"I did some good things," he said. "Nothing special. Nothing terrible."
Copyright 2015 by The Associated Press