Minev came to Levski in June 2006. On 9 May 2009, Minev and his twin brother Yordan entered in the history of The Eternal Derby. The situation was even more interesting, because Veselin was a left full-back and his brother Yordan was a right midfielder, and Veselin guarded his brother Yordan. Despite the tackles, after a foul against Veselin, Yordan came to him and call the referee. He became a Champion of Bulgaria in 2008/2009 season. In 2009/2010 season, Levski achieved qualifying for UEFA Europa League becoming 3rd in the final ranking. During the 2010/2011 Levski qualified for UEFA Europa League after eliminating Dundalk F.C., Kalmar FF and AIK Fotboll. Levski was drawn in Group C, facing Gent, Lille and Sporting CP.
On 30 May 2011, Minev signed a three-year deal with Turkish club Antalyaspor for an undisclosed fee. He left the club in early December 2012 for personal reasons.
Botev Plovdiv
On 12 December 2012, immediately following the appointment of Stanimir Stoilov as head coach, Minev signed a 1.5-year contract with his former club Botev Plovdiv. Minev was among the first choice defenders during his time with the "canaries", captaining the team on occasions. His last game for the club was the 4:0 home victory against Libertas in the first qualifying round for UEFA Europa League on 3 July 2014.
Levski Sofia
In July 2014 Minev's contract with Botev was terminated due to the financial problems that occurred in the club. On the 7 July 2014 he returned to Levski signing a new 1+1 year deal with the club. He was released in June 2017.
Etar
On 16 June 2017, Minev signed a 1-year contract with newly promoted Etar Veliko Tarnovo.
On 11 June 2018, Minev signed a 1-year contract with Second League club Tsarsko Selo.
International career
On 28 August 2009, Minev received his first international call-up for the World Qualifier match against Montenegro. He remained an unused substitute for the game. On 10 October 2009, he made his debut for the national side in the 1:4 away loss to Cyprus, playing the full 90 minutes of the game. Minev was a starter for Bulgaria during the first two 2014 World Cup qualifiers, but in October 2012 he received a two-match ban for his involvement in a number of incidents during the ill-tempered match against Armenia.
