Super Bowl champions New Orleans picked 32nd, taking Florida State corner back Patrick Robinson.
The opening round of the seven-round draft took place on prime-time TV for the first time.
The three tiers of balconies in New York's Radio City Music Hall were packed with fans of the 32 NFL teams cheering and jeering the selections.
The second and third rounds of the draft are held on Friday, with rounds four to seven taking place on Saturday. The new season kicks off on Thursday 9 September, with the Saints playing host to the Minnesota Vikings in a rematch of last season's NFC Championship game.
Following three rounds of playoff action, Super Bowl XLV is played at Dallas Cowboys Stadium in Texas on 6 February.
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