Nieto grew up in Long Beach California, in a dangerous neighborhood; his mother had stated, "There were drive-by shootings. He jokes about it, that Snoop Dogg lived around the corner. It's true, but Matt wasn't born then." When he was two years old, Nieto became interested in skating, particularly rollerblading, when he saw his sister skating in the family's house. His grandfather later bought him a hockey stick, and when he was three, Nieto began playing roller hockey at the YMCA, and was a member of the L.A. Hockey Club, being teammates with future NHL players Emerson Etem and Rocco Grimaldi.
Nieto began attending Salisbury School, a preparatory school, in Salisbury, Connecticut, at the start of his sophomore year of high school. In one season with the school's hockey team, he scored eight goals and ten assists in 28 games. The performance drew the attention of the USA Hockey National Team Development Program, and Nieto subsequently moved to Ann Arbor, Michigan, to play in the Program. In 2008–09, he played for the under-17 and under-18 teams, scoring 20 goals and 33 assists and six goals and eight assists, respectively. The next season, he played for the Program's United States Hockey League (USHL) team, scoring 15 goals and 14 assists; he also returned to the under-17 team, scoring one goal.
After participating in the Program, Nieto was offered a chance to play major junior hockey for the Tri-City Americans of the Western Hockey League (WHL), but opted to attend college instead and play at the collegiate level. When choosing colleges, Nieto decided between Boston University and Boston College, and elected to attend the former on a full scholarship. In his freshman year, Nieto scored 10 goals and 13 assists. The next year, he scored 16 goals and 26 assists with a plus-minus rating of +19. In 2012, Nieto scored a 18 goals, second-most on the team behind Dan O'Regan, and 19 assists. He ended his Terriers career with 44 goals and 102 points in 115 games.
Nieto was ranked the 43rd-best prospect and the second-best player in the NCAA's Hockey East conference by NHL.com among the 2011 draft class. Hockey's Future stated he had struggled earlier in his career defensively, but had since improved. They added he had "explosive speed," and excels with puck handling and passing while moving. The website gave him a 7.5/10 talent score.
In the Draft's second round, Nieto was selected by the San Jose Sharks with the 47th overall pick, their first pick of the draft. Nieto became the first Californian to be selected by the team. Regarding Nieto, Sharks Director of Scouting Tim Burke stated, "We were watching this guy over two years now. He has offensive ability on the rush and has shown he can do other roles, like penalty kill."
Nieto signed a three-year, $2.3 million contract with the Sharks on April 1, 2013. He spent time with the Sharks' then-American Hockey League (AHL) affiliate, the Worcester Sharks, during the year, playing in 11 games. He scored two goals with four assists, a –4 plus-minus rating and six penalty minutes.
Nieto made his NHL debut against the Vancouver Canucks on October 3, 2013. He scored his first career NHL goal against Henrik Lundqvist of the New York Rangers on October 8, off assists by Joe Pavelski and Tommy Wingels. Nieto, however, returned to Worcester over one month later, on November 25, though he was later recalled on December 7. Despite being reassigned to the San Francisco Bulls of the ECHL on December 18, Nieto was recalled by the Sharks three days later after an injury to forward Tomáš Hertl.
Nieto scored his first career Stanley Cup playoff goal on April 22, 2014, via an empty net goal against the Los Angeles Kings.
In the 2016–17 season, Nieto struggled to retain his role in adding an offensive touch to the Shark's bottom six forwards. Surpassed in the depth chart, Nieto played in just 16 games with 2 assists before he was placed on waivers in order to gain more playing time. Nieto was claimed off waivers by the Colorado Avalanche the following day, on January 5, 2017.
Nieto was first selected to represent the United States at the 2009 IIHF World U18 Championships, posting four goals in seven games to help capture the gold medal. He was reselected the following year and successfully defended Team USA's gold medal at the 2010 IIHF World U18 Championships.
Nieto is of Mexican-American descent. Nieto's father, Jesse, is a longshoreman, while his mother, Mary, is a Nordstrom makeup artist.