Claiborne said that he had never spoken to Cowboys executives before they traded up eight spots Thursday night to take the All-American who was considered by most as the best defensive back available.
"I didn't see it coming in a million years. I had no idea," Claiborne said. "We didn't talk none, none at all. That's why it was shocking to me when I got the phone call."
While the Cowboys went into the draft in need of defensive playmakers, Jones said Claiborne was the only player they were willing to make such a significant deal to get.
"Use the word elite, we think he is," Jones said. "I think he's the most consensus pick and the most consensus move I can ever remember on our draft day."
This was the 59th draft-day trade for Jones since he bought the team in 1989. It was the 18th time one of those deals involved his first-round pick.
Claiborne led the Tigers with six interceptions last season and won the Jim Thorpe Award as the nation's top defensive back. He had 11 picks in 26 games the last two seasons. He was also LSU's top kickoff returner, averaging 25 yards per return including a 99-yard touchdown at West Virginia.
"I know they sacrificed to come down to get me, but I feel like I'm worth it," Claiborne said. "I know my talents, I know what I'm able to do and feel like I can come in right away and get with the team, learn the system and compete."
Claiborne was asked about reports that he scored poorly on the Wonderlic test, which is designed to gauge a prospect's intelligence, problem-solving ability and cognitive skills.
"I know what type of person I am, I know that tests don't reflect how I learn or what type of person I am," Claiborne said. "I looked at the test, no questions that came with football, so I pretty much blew the test off. ... I'm here for football."
Jones said the test results were "just not an issue with us at all" and that the Cowboys understand Claiborne's "ability to play the game and play the defense." Jones said those scores don't take into account how the player takes coaching, or his football instincts and skills.
Among people Dallas spoke with about Claiborne was LSU coach Les Miles, a former assistant with the Cowboys.
Claiborne is recovering from surgery done after the NFL combine to repair ligament damage in his left wrist.
The Cowboys don't consider the injury serious even though he is wearing a cast and then will have to wear a brace, keeping him out of action during the rookie minicamp next week and other organized team activities in May and June. But they expected him to be ready for the start of training camp in late July.
Now they have used their highest draft pick since 2003 - when cornerback Terence Newman was picked fifth overall - on another top corner.
Claiborne is from Shreveport, La., about a three-hour drive from Dallas. Being unexpectedly picked by the Cowboys was a nice surprise for his family.