Houston traded its 2017 first-round pick (No. 25) and 2018 first-round pick to move up to No. 12.
Combined with an earlier first-round trade between the teams, the Texans traded two first-round picks (one from 2018), a second-round pick ('18), a sixth-round pick and Brock Osweiler to the Browns for Watson and a fourth-round pick.
Watson called the selection a "perfect situation" for him after he was selected. Texans coach Bill O'Brien and general manager Rick Smith said in conference calls that Tom Savage would remain the starting quarterback, a point they reiterated Thursday night, and that Watson would come in and learn.
"As a rookie quarterback it's a big jump," O'Brien said. "Tom is our starter and Deshaun will come in and he's going to work hard and we're going to teach him and feed him a lot of information, and he'll work at it."
Said Smith: "I don't know if people believe us but we're comfortable with Tom Savage as our quarterback."
Watson was the third quarterback taken, nearly five months after leading Clemson to a national championship, with North Carolina's Mitchell Trubisky going to the Chicago Bears at No. 2 and Texas Tech's Patrick Mahomes to the Kansas City Chiefs at No. 10.
"It's amazing," Watson said on the ESPN broadcast. "I tell myself not to cry but can't hold it back. Now I'm finally being introduced into the National Football League. It's a blessing, but I worked for it."
Watson brought his mother, Deann, onto the draft stage after the selection.
"She's the one who raised me," said Watson, who will be formally introduced by the Texans at a news conference Friday at 3 p.m. ET. "And I don't even know what to say. I'm going to take care of her for sure."
Earlier this month, Texans star receiver DeAndre Hopkins said he was biased because he is also "a Clemson guy," but that he's "pretty sure [Watson] would like to throw me the football."
Watson is the first quarterback the Texans have taken in the first round since they selected David Carr in 2002.
Last season at Clemson, Watson threw for 4,593 yards and 41 touchdowns, leading Clemson to a national title.
Information from ESPN's Dan Graziano was used in this report.