Media playback is not supported on this device
Tunsil had been projected to be the first man taken in the Draft, where NFL teams sign the best college players.
But, after trading with the Tennessee Titans, the Los Angeles Rams used first pick to take quarterback Jared Goff.
The Miami Dolphins signed Tunsil, who said his Twitter account was hacked.
The 21-year-old admitted the person in the video, posted moments before the draft, was him but said the incident occurred "several years ago".
"I made a mistake, a huge mistake," said the highly rated offensive lineman, who played for the University of Mississippi.
However Tunsil said he does not have a drug problem, adding: "You can check all my college tests. I never failed one."
Dolphins general manager Chris Grier said the team were "very comfortable" with the information they had on Tunsil, having completed background checks.
CBSSports.com national columnist Bill Reiter said the situation threatened to overshadow the draft.
"The internet - one of the few things bigger than the NFL - hijacked the NFL draft on Thursday, turning a celebration of the game into a real-time soap opera involving drug use, accusations of pay-for-play shenanigans, the freefall of the player involved, and one bizarre press conference," he wrote.
Media playback is not supported on this device
The Los Angeles Rams, who have relocated from St Louis, traded with the Tennessee Titans to secure the top pick.
They gave up six picks across the top three rounds this year and in 2017 to allow them to choose Goff.
The 21-year-old, from California University, was chosen ahead of fellow quarterback Carson Wentz, who was taken by the Philadelphia Eagles.
The San Diego Chargers selected defensive end Joey Bosa with the third pick, while the Dallas Cowboys picked up Ohio State back Ezekiel Elliott.
The New England Patriots were without their first-round pick as a punishment for the 'deflate-gate' scandal.
Rounds two and three of the draft, held at Chicago's Roosevelt University, take place on Friday, with the last four rounds on Saturday.
Media playback is not supported on this device
College star Laremy Tunsil was picked 13th in the NFL Draft - after a video appeared on his Twitter account seemingly showing him smoking marijuana while wearing a gas mask.