He played college football at SMU and was selected by the New Orleans Saints in the fifth round of the 2009 NFL draft.

Morstead attended Pearland High School and won varsity letters in football and basketball.

[3][4] In football, Morstead received second-team All-District honors and was named the Brazoria County Special Teams MVP as a senior.

[8] Morstead took over place-kicking and punting chores in 2006, earning All-Conference USA third-team honors.

[10] As a sophomore, Morstead was a consensus All-Conference USA first-team pick and also gained league academic honors.

[20] Morstead played a role in the Saints' victory over the Indianapolis Colts in Super Bowl XLIV by executing an onside kick during the second half kickoff.

[23] Morstead currently holds the record for most kickoff touchbacks in one game with nine, which he set in a 62–7 victory over the Colts on October 23, 2011.

[26] In July 2012, the Saints signed Morstead to a six-year extension stated to be worth $21.9 million, making him the second highest paid punter in the league (after Shane Lechler of the Oakland Raiders).

[25] He went on to have an outstanding season, leading the league (with a record-setting pace through 15 games) in net punting yardage,[27] and was elected to the Pro Bowl.

[36] In the NFC Divisional Round against the Minnesota Vikings, he tore cartilage in his rib cage after making a tackle in the first quarter but remained in the game.

[37] The Vikings scored the winning touchdown in the closing seconds of the game, and both sides assumed that the contest was over with Saints players headed for the locker room.

[38][39] Vikings' fans were impressed by the toughness and sportsmanship Morstead displayed in the eventual defeat, so a Vikings-dedicated Reddit group donated more than $140,000 to his charity in less than 24 hours, and Morstead presented the donations to the Children's Hospital of Minnesota.

[49] On September 14, 2021, Morstead signed with the New York Jets[50] to fill in for the injured Braden Mann.

[53] Morstead was also later named NFC Special Teams Player of the Month for December.

[56] During a narrow Week 3 21–19 victory over the Buffalo Bills, Morstead inadvertently kicked a punt from the endzone into the buttocks of teammate Trent Sherfield, resulting in a safety.