Jones told reporters the Cowboys received three "worthwhile trade offers" for the No. 17 pick but decided to stay in the spot for the Oklahoma star.
There seemed to be a collective surprise between Lamb and the Cowboys that he was still available when they made their first pick.
Lamb told reporters afterward he was "very surprised" that Dallas drafted him but added he was "thrilled" to join the organization.
Most mock drafts projected Lamb to come off the board in the first half of the first round. Bleacher Report's Matt Miller projected him to be the first wide receiver taken by the Las Vegas Raiders with the 12th pick.
Leading up to the draft, the Cowboys seemed likely to target defense early after losing Byron Jones, Robert Quinn, Maliek Collins and Jeff Heath in free agency.
As Jones said, the fact that Lamb was still on the board "made the decision" for the Cowboys, and they "just didn't want to miss it."
Jori Epstein of USA Today noted Dallas' front office is aware of the need to add defensive backs and defensive linemen but feels depth at those positions will allow the team to address those spots on Day 2 of the draft.
Even with those holes on defense, the Cowboys boast an offense led by Dak Prescott with skill position players like Lamb, Amari Cooper, Michael Gallup and Ezekiel Elliott to create problems for opposing teams next season.