Berjón scored his first professional goal on 13 September 2008, in a 2–2 home draw with Real Zaragoza.[3] He added four more during his debut campaign with the Canary Islands side.
On 19 August 2010, Berjón joined FC Barcelona for a €300,000 fee,[4] being assigned to the reserves also in the second division.[5] He appeared regularly for the Catalans, before being loaned to AD Alcorcón in the same league on 30 August 2011.[6]
On 4 August 2012, Berjón signed for Real Murcia.[7] After narrowly avoiding relegation in 2012–13[8] he featured in 43 matches the following season, again scoring six times and providing a league-best 20 assists as the Pimentoneros finished fourth but were relegated due to financial problems.[9]
On 26 August 2014, Berjón agreed to a two-year deal at SD Eibar, newly promoted to La Liga.[10] He made his debut in the competition four days later, replacing Dani García in a 2–1 defeat at Atlético Madrid.[11] He scored his first league goal on 4 September, through a spectacular volley in a 3–3 home draw against Levante UD.[12]
Berjón moved abroad for the first time in his career on 24 June 2016, signing for Mexico's Club Universidad Nacional.[13] On 28 December, he returned to Spain and its second division after joining his first club Oviedo.[14]
On 15 September 2020, the 34-year-old Berjón signed for third-tier Burgos CF.[15] They were promoted at the end of his first season, but he left on 9 January 2023 after falling down the pecking order.[16]