Jenkins was selected No. 41 overall by the Redskins, and ahead of former Tiger defensive end Da'Quan Bowers.
Jenkins was a first team All-ACC defensive tackle last year when he had 53 tackles, including nine tackles for loss.
Gilchrist was selected by San Diego with the No. 50 selection.
The North Carolina native had 66 tackles to rank fourth on the team in 2010 and he also led the Tigers in punt returns and kickoff returns.
Da'Quan Bowers, originally thought of as a first round lock, was drafted after Gilchrist with the No. 51 overall pick (Tampa Bay).
Bowers led the nation in sacks with 15.5 and was second in tackles for loss with 26 in 2010.
He was also the recipient of the Bronko Nagurski Award as the National Defensive Player of the Year by the Football Writers Association, and winner of the Ted Hendricks Award as the top defensive end in college football.
The Bucs also drafted former Clemson defensive end the late Gaines Adams four years ago with the No. 4 overall pick.
As many as two other former Tigers could be selected either in the second or third round of tonight's draft.
Jenkins, Gilchrist, Bowers drafted
Panther Insider Top Stories
Will the Panthers snap out of their funk?Video: What's wrong with the Carolina Panthers and what it will take for them to turn their season around after starting 1-4.
Scout NFL Network10/12/2016
Raiders find a way to win, move to 9-2The Oakland Raiders defeated the Carolina Panthers on Sunday to move to 9-2 on the season
SB ReportSunday at 2:47 PM
Countdown to Paydirt against CarolinaA look at the matchup between the Oakland Raiders and Carolina Panthers
Week 9: DFS Pivots Plays Vs. The ChalkFantasy Football Expert Armando Marsal discusses Week 9 pivots to cash BIG when playing NFL DFS on DraftKings!