Born in Port Kembla, New South Wales, Cooper originally gained an interest in rugby league while watching and supporting Sydney's Western Suburbs Magpies, the club his father followed. He began playing rugby league at the age of six for his local junior side, the Shellharbour Sharks, while also enjoying athletics and being actively involved with his local surf club with his close cousin Corey.
After moving to Illawarra, Cooper began supporting the Illawarra Steelers. Cooper was originally signed on to the Illawarra Steelers club as a teenager prior to their merger with the St. George Dragons at the end of 1998. He made his debut for the merged St George Illawarra club in 2000.
After the merger of the St George Dragons and the Illawarra Steelers clubs in 1999, Cooper was signed on with the new St. George Illawarra Dragons team, quickly impressing club officials and at the age of twenty-one being awarded his NRL début in the opening round of the 2000 season against local rivals the Cronulla-Sutherland Sharks, he had an uneventful début without managing to score.
After making his début Cooper was replaced in the Dragons line-up for the second game of the season and was not re-called until round seven in which he crossed the line for his first try against the New Zealand Warriors, Cooper in total scored two tries in limited appearances in the 2000 season.
The 2001 season was not a successful one for Cooper, with minor injuries causing him to miss many games. He returned just before the finals, scoring two tries in the semi-final victory over the Bulldogs.
In 2002, Cooper was able to retain his position in the team for the majority of the season and finished the year with a total of seven tries from nineteen games.
Cooper held down his spot for the duration of the 2003 season. In total, Cooper scored eleven tries including three doubles during the regular season, he then tasted his first representative honour being selected to play for Country in the annual City and Country Origin. Although the season eventually finished on a low note both for the Dragons and Cooper with another injury yet again forcing him out of the side. Though all was not lost for Matt as he was nominated for the Dragons player of the year, eventually losing out to Ben Hornby while he picked up the Geoff Selby Memorial NRL Coaches Award.
The 2004 season yet again was one of improvement and new opportunity, Cooper began the season well scoring a total of eight tries in six matches for the Dragons including four tries in a single match against the reigning premiers the Penrith Panthers. Later Cooper was called up for the third and deciding game of the 2004 State of Origin series for New South Wales after the side had lost the second match, replacing Luke Lewis to partner his clubmate Mark Gasnier in the centres.
At the conclusion of the 2004 St. George Illawarra Dragons season Gasnier had scored the most tries, with a total of seventeen, and the international selectors assigned him a spot in the Australian side for the 2004 Kangaroo tour of Great Britain, France and the United States. His club coach Nathan Brown had recommended that Cooper be selected for the Australian rugby league side under coach Wayne Bennett, though he eventually opted for the usual more experienced international representatives and Cooper did not play a competitive match on the tour.
After having an impressive début match in the previous series, Cooper was subsequently re-selected for New South Wales in the 2005 State of Origin series, playing in all three matches of the series helping New South Wales to another series win, scoring one try along the way in game two. He finished the NRL season well with thirteen tries from twenty games and was again selected to go on to tour of Great Britain with the Australian national side. He appeared in the first three test matches on the tour, scoring five tries.
Injury yet again hit Cooper early on in the 2006 season in the form of a damaged hamstring while playing in the season opening fixture against the Wests Tigers. Cooper was forced to miss several fixtures for the Dragons after failing to recover from a hamstring injury sustained during the round one NRL fixture. However Cooper was once again selected to play for Australia in the mid-season test match on 5 May 2006, he scored once during a dominating 50 to 12 win over New Zealand. This was the last time Cooper has represented Australia. His hamstring injury forced Cooper to withdraw from the starting line-up for the 2006 State of Origin series, being replaced by winger Eric Grothe, Jr. from the Parramatta Eels. After missing the first game of Origin for New South Wales, Cooper failed to be called up for game two with Grothe retaining his position after a solid performance in game one. On 11 June it was announced that Cooper had been called up as 18th man on stand by for the game.
In August 2008, Cooper was named in the preliminary 46-man Kangaroos squad for the 2008 Rugby League World Cup.
He was selected for Country in the City vs Country match on 8 May 2009. However, he did not play in the match after withdrawing with injury.
In the first round of the 2010 finals, St George Illawarra Dragons defeated the Manly-Warringah Sea Eagles 28-0, with Cooper scoring two tries, surpassing Nathan Blacklock's record of 100 tries to become the Dragons highest try scorer, with 102 tries. In 2010 he got a recall to the Origin side. He played Games 1 and 2 but sustained an injury which kept him out of Game 3. On 3 October 2010, Cooper was a part of the Dragons side that beat the Sydney Roosters 32-8 in the 2010 NRL Grand Final, helping the joint venture club claim its first premiership. This was also Cooper's 100th career win out of a possible 199 NRL games.
The following February Cooper travelled with the Dragons to England to play the Super League champions, Wigan in the 2011 World Club Challenge. He scored a try in St George Illawarra's victory. Cooper ended the 2011 season as St. George Illawarra's leading tryscorer with 14.
Cooper was a participant in the fourteenth season of Dancing with the Stars.
In 2015, Cooper is a contestant in the fourth season of The Celebrity Apprentice Australia.