Griffin said he found out about the decision from Jackson on Monday morning in a meeting with all of the quarterbacks.
"It's an honor," he said. "It's an honor to be one of 32 that are in this league. You take that and you respect that and you continue to go out and work and try to get better."
Griffin, who had been battling Josh McCown for the starting job, becomes the 25th starting quarterback for the Browns since the franchise returned to the NFL in 1999. Brian Hoyer is the only one of those quarterbacks to have a record over .500, according to ESPN Stats & Information.
"He played the way I think a quarterback has to play. I think he did it flawlessly. I think he did everything I asked him to do. I know sometimes you guys judge him maybe a little different than I do; I get that. But I think that he did a great job and he's continuing to do a great job. But again, he's got to continue to get better, too," Browns coach Hue Jackson said.
Griffin was rookie of the year and made the Pro Bowl in his rookie season after being selected No. 2 overall in the 2012 draft by the Washington Redskins. A series of injuries, however, derailed his career, and last season he lost his starting job in Washington to Kirk Cousins.
The two sides moved on after the season, and Griffin signed a two-year, $15 million contract with the Browns, who went 3-13 last season and haven't had a winning campaign since 2007.
Griffin said he never doubted he would get another chance.
"Everything that you go through in life, is a lesson," he said, "and it's up to you to decide to use that lesson in a positive or negative way."
Jackson said Griffin has improved every day and should get even better as "he becomes more and more acclimated to the offensive system we will employ this season.
"I've been crystal clear on what we expect from Robert," Jackson said. "He fully understands the responsibility, and I feel he is ready to handle this role."
Teammates have been supportive of Griffin throughout camp.
"He's worked his tail off," cornerback Joe Haden said. "He's not done anything wrong. He's been a really good leader. He's been working. He's in control of the offense."
Terrelle Pryor said Griffin earned the job. In the offseason, Pryor posted a photo on Instagram of Griffin sitting at his locker studying the playbook before Pryor arrived for work at 7 a.m.
"That's how he is every day," Pryor said.
McCown started eight games last season but battled injuries, opening the door for Johnny Manziel. Off-the-field problems ultimately derailed Manziel's career; he remains unsigned.
McCown said it was better for the Browns to decide sooner rather than later who would start.
"I've been on both ends of it, and when quarterback competitions ... they drag out through training camp," he said. "I don't know if you're any better off because then you have two guys having half [the] reps. I think it's great that we know who it is moving forward. We're behind that guy, and our team can rally around that guy."
ESPN's Pat McManamon contributed to this report.