After departing the youth setup at
Chelsea in May 2011, Rodgers joined the academy of recently promoted Championship club
Brighton & Hove Albion, signing an initial one-year contract with the option of a further year which was taken up by the club in the summer of 2012. Rodgers made his professional debut and only appearance for Brighton in an
FA Cup match against
Wrexham on 7 January 2012. On 20 March 2013, Rodgers joined
League Two side
Exeter City on an initial two-month loan deal. Rodgers made his professional League début for Exeter on 9 April during a 4–1 defeat at
Rotherham United, appearing as a 70th-minute substitute. On 21 May 2013, Rodgers was released by Brighton, along with former Chelsea teammate
Ben Sampayo. On 12 July 2013, Rodgers signed for
League One side
Oldham Athletic on a one-year contract. On 10 August 2013, Rodgers made his Oldham debut in their 1–0 home defeat against
Walsall, replacing
James Dayton in the 74th minute. On 3 September 2013, Rodgers was given his first Oldham start in their
Football League Trophy away tie against
Shrewsbury Town, featuring for 71 minutes before being replaced by
David Mellor. The game resulted in a 4–1 victory for the visitors with
Danny Philliskirk scoring a hat-trick. Rodgers was released by Oldham at the end of the 2013–14 season after limited appearances in the first team. On 17 July 2014, preceding his release from Oldham, Rodgers joined League One side
Swindon Town. On 23 September 2014, Rodgers made his Swindon debut in a Football League Trophy 2–1 victory over
Newport County, in which he replaced
Louis Thompson in the 58th minute. A month later, Rodgers scored on his league debut for the club in their 3–2 home defeat against
Rochdale, netting the game's opener in the 35th minute. In June 2017, Celtic manager
Brendan Rodgers fielded his son, Anton, as a trialist during the Scottish champions' 1–0 friendly win over
BW Linz in Austria. He was a free agent after leaving Swindon Town. On 3 November 2017, Rodgers joined
National League South side
Hungerford Town following his release from Swindon. Following a lack of game time, Rodgers opted to leave Hungerford; however, just two months later he rejoined the club. In July 2018, Rodgers was fielded as a trialist for
Peterborough United in their pre-season friendlies. In September 2018, he joined
Falkirk on trial after leaving Hungerford at the end of the 2017/18 season. Rodgers featured for
Wealdstone and
Bracknell Town during the 2018–19 season. On 19 August 2019, Rodgers joined
Marlow. On 1 October 2021,
Basingstoke Town announced the signing of Rodgers for the 2021-22 season. ==International career==