Always considered to be one of the best players in his age group in Europe, Vezenkov began playing youth club basketball in Cyprus in 2005 with APOEL Nicosia's youth teams. In 2009, he moved to Greece and began playing youth club basketball with Aris' youth teams, until 2011, when he joined Aris' senior men's club. He trained for several years at Nick Galis Hall, located near the center of Thessaloniki, Greece.
On 16 May 2013, Vezenkov committed to play American college basketball in the Big East Conference, at Xavier University, with the Xavier Musketeers. However, he ultimately decided to stay in Greece with his club team Aris instead.
Vezenkov began his pro career with the senior men's team of Aris, in the Greek League, in 2011, appearing in 10 games, with an average of 2.5 minutes on the floor. In the 2012–13 season, he appeared in 21 games, averaging 3.9 points and 2.5 rebounds per game. He was named the Greek League Best Young Player in 2013.
In August 2013, he re-signed with Aris for 3 years. In the next season, he established himself in the team, appeared in 26 games and averaged 11.1 points, 5.8 rebounds, 2.1 assists, and 0.9 steals per game. He was named the Greek League Best Young Player again in 2014. In the local derby against rivals PAOK, during the 2014–15 season, Vezenkov achieved Greek League career-highs in scoring and rebounding, with 29 points and 14 rebounds respectively. In total, he played in 37 games during the 2014–15 Greek Basket League season, averaging 16.8 points, 7.3 rebounds, and 1.6 assists per game. During 26 regular season games, Vezenkov led the league in total points scored, with 469, and was second in total rebounds grabbed, with 201. He was also named the league's MVP, Most Improved Player, and Best Young Player that season. Vezenkov had originally planned to enter the 2015 NBA draft, but later withdrew his name from it before the deadline that was set for international players back in June.
On July 31, 2015, Vezenkov signed a four-year deal worth €2 million euros net income, with the Spanish ACB League club FC Barcelona. FC Barcelona also paid a €315,000 euros buyout fee to Vezenkov's former club, Aris. Vezenkov's contract with FC Barcelona also included a buyout option for the NBA, set at the amount of €1 million euros. The Spanish League club Valencia Basket, and the Greek League club Panathinaikos, also were interested in signing Vezenkov to a contract at the time. Vezenkov would then put his name in the 2016 NBA draft, after his first season with Barcelona ended. However, he wound up removing his name from the draft on the June 13, 2016, deadline for international players.
With Barcelona, Vezenkov won the Spanish Supercup title in 2015, and was named to the all Spanish League All-Young Players Team of the 2016–17 season.
On June 22, 2017, Vezenkov was selected in the 2nd round, as the 57th pick overall of the 2017 NBA draft, by the Brooklyn Nets.
Vezenkov was a member of the Bulgarian junior national teams. With Bulgaria's junior national teams, he played at the following tournaments: the 2010 FIBA Europe Under-16 Championship, the 2011 FIBA Europe Under-16 Championship, the 2012 FIBA Europe Under-18 Championship, the 2012 FIBA Europe Under-20 Division B Championship, and the 2013 FIBA Europe Under-18 Championship. He led the 2011 FIBA Europe Under-16 Championship in scoring, with an average of 27.1 points per game.
He led the 2013 FIBA Europe Under-18 Championship in scoring, with an average of 22.4 points per game. He also led the 2014 FIBA Europe Under-20 Championship in scoring, averaging 19.3 points per game, and rebounds, averaging 11.2 rebounds per game. He was also named to the tournament's All-Tournament Team.
Vezenkov also represented the senior men's Bulgarian national team in the second qualification tournament for FIBA EuroBasket 2015, averaging 17.3 points per game, and also being the highest scoring player for the team, in 5 of their 6 games, as well as the team's overall top scorer. He also played at the EuroBasket 2017 qualification.
Note: The EuroLeague is not the only competition in which the player participated for the team during the season. He also played in domestic competition, and regional competition if applicable.3x Greek League Best Young Player: (2013, 2014, 2015)
Greek League Most Improved Player: (2015)
Greek League Best Five: (2015)
Greek League Top Scorer: (2015)
Greek League MVP: (2015)
Spanish Supercup Winner: (2015)
Spanish League All-Young Players Team: (2017)
2011 FIBA Europe Under-16 Championship: Top Scorer
2013 FIBA Europe Under-18 Championship: Top Scorer
2014 FIBA Europe Under-20 Championship: Top Scorer
2014 FIBA Europe Under-20 Championship: All-Tournament Team
2x Best Young Basketball Player of Bulgaria: 2013–14, 2014–15
2nd Place in the Viasport.bg "Best Young Sportsperson of Bulgaria" rankings: 2014
Vezenkov is the son of Ianka Vezenkova (née Gerginova) and Sachko "Sasho" Vezenkov, who has served as both head coach and director of sports of Lukoil Academic. Sasho is also a former international basketball player, who played at EuroBasket 1985, EuroBasket 1989, and EuroBasket 1991. Sasho was also the captain of the senior men's Bulgarian national basketball team. Sasho moved to Cyprus, and he played and coached basketball there in the Cypriot Basketball League for many years.
Vezenkov's older sister, Michaela Vezenkova, played college basketball at UNC Wilmington, and also played professionally in the Cypriot women's league. Vezenkova decided to represent Cyprus, rather than Bulgaria, and she eventually became the captain of the Cypriot women's national basketball team.
While living in Greece as a youth, Vezenkov attended Greek schools. Vezenkov is fluent in both the Greek and the Bulgarian languages, with Greek being his native tongue, and Bulgarian being his second language.
Vezenkov officially received Greek citizenship on December 30, 2015, under the official name of Alexander "Alex" Vezenkof (Greek: Αλεξάντερ "Αλέξ" Βεζένκοφ), with his common Greek name being Sasha Vezenkof (Greek: Σάσα Βεζένκοφ). Before acquiring Greek citizenship, Vezenkov had previously stated that he felt like a member of Greek society, since he was born and raised in Greek communities in both Cyprus and Greece.