The pick was announced by Browns great Jim Brown, the franchise's all-time leading rusher.
Chubb starred in the backfield for Georgia, rushing for 1,345 yards and 15 touchdowns in 2017.
In four seasons at Georgia, he rushed for 4,769 yards (second all time in SEC history) and 44 touchdowns. He also had 31 receptions and four touchdown catches.
"I've been a big fan of Chubb for a long time. One thing about SEC running backs is when you play running back in the SEC you have to create between the tackles with your feet. You have to be able to play a physical style of football," Browns vice president of player personnel Alonzo Highsmith said. "And Nick Chubb exemplifies all of that in his running style."
"I have no questions about the guy. Just give him the ball."
The Browns opened the second round by selecting guard Austin Corbett from Nevada with the 33rd overall pick. Corbett could be the long-term replacement at left tackle, filling the void left by Joe Thomas, who retired after 10 season this offseason.
Chubb is the second Georgia running back drafted in 2018, as Sony Michel was taken by the New England Patriots in the first round on Thurdsay. Georgia is the first school with multiple running backs taken within the first two rounds of the same draft since Arkansas in 2008 when Darren McFadden and Felix Jones were selected in the first round.
ESPN's Pat McManamon and ESPN Stats & Information contributed to this report.