The Texans, picking first after a league-worst 2-14 season, kicked off the annual selection party at Radio City Music Hall by taking defensive end Clowney to partner with two-time All-Pro DE J.J. Watt.
"I'm going to do everything I can to help the program," said Clowney, who registered 24 sacks and 130 tackles in three seasons and became the first sophomore to win the Hendricks Award as the nation's most outstanding defensive end.
Commissioner Roger Goodell was subjected to some jeering as he officially opened the draft, which produces the primary pool of talent for NFL teams as well as plenty to cheer and boo for thousands of fans in attendance decked out in team jerseys.
Clowney was the first of 32 picks in the first round of the draft. Rounds two and three will be held on Friday with the last four rounds conducted on Saturday in the three-day extravaganza.
Clowney, who stands 6-feet-5 (1.96 m) and weighs 266 pounds (121 kg), stunned observers at the NFL Scouting Combine in February by running a blistering 4.53 seconds in the 40-yard dash - a time better than many of the wide receivers tested.
The St. Louis Rams used the second pick of the draft to take Auburn offensive tackle Greg Robinson, with the Jacksonville Jaguars grabbing quarterback Blake Bortles of Central Florida University with the third choice.
Jacksonville's selection of Bortles stunned the crowd, many of whom had anticipated that Texas A&M's Johnny Manziel would be the first quarterback taken in the draft.