Marsh is the son of Curtis Marsh Sr. and the half-brother of Curtis Marsh Jr. Marsh’s father, a wide receiver who played college football at Utah, was drafted in the seventh round of the 1995 NFL Draft by the Jacksonville Jaguars. Curtis Sr. played two seasons for the Jaguars and one for the Pittsburgh Steelers before going to the Canadian Football League, where he was an All-Star in 2000. Curtis Jr. played college football at Utah State; he was a running back for the 2007 and 2008 seasons before moving to defensive back for the 2009 and 2010 seasons. Curtis Jr. was selected in the third round of the 2011 NFL Draft by the Philadelphia Eagles.

In December 2010, he was selected to the Rivals.com Pac-10 All-Freshman team in his freshman season. He was winner of UCLA’s John Boncheff Jr. Memorial Award for Rookie of the Year in his freshman season.

Marsh was drafted by the Seattle Seahawks in the fourth round, 108th overall, in the 2014 NFL Draft. He was signed to a four-year contract on May 19, 2014. Marsh recorded the first sack and first forced fumble of his NFL career against the Miami Dolphins in the season opener on September 11, 2016.

On September 2, 2017, the Seahawks traded Marsh to the New England Patriots in exchange for the Patriots’ fifth- and seventh-round selections in the 2018 NFL Draft. On October 22, he blocked a field goal in a game versus the Atlanta Falcons. On November 21 the Patriots waived Marsh.

Marsh was claimed off waivers by the San Francisco 49ers a day after being released by the Patriots. On February 7, 2018, Marsh signed a two-year contract extension with the 49ers.

In 2018, Marsh played in 16 games with three starts, recording a career-high 38 combined tackles and 5.5 sacks.

On March 15, 2019, Marsh was released by the 49ers.

On April 4, 2019, Marsh signed with the Seattle Seahawks. He was released on August 31, 2019.

On September 1, 2019, Marsh signed with the Arizona Cardinals.

Marsh signed with the Jacksonville Jaguars on April 6, 2020. He was released on October 15, 2020.

On October 24, 2020, Marsh was signed to the Indianapolis Colts practice squad. On November 12, 2020, he was promoted to the active roster.