Veselý began playing basketball with the youth clubs of Příbor and BK Snakes Ostrava in the Czech Republic. He joined Geoplin Slovan in 2007, he then moved to KK Partizan in 2008. He won nine club trophies with the club, and also reached the 2009-2010 Euroleague Final Four In 2010, Veselý was presented the FIBA Europe Young Men's Player of the Year Award. At his last match with Partizan, the final of the 2010-11 Serbian League, the team captain Petar Bozic let Veselý lift the winning trophy, while the audience gave him a standing ovation.

Veselý was drafted in the first round by the Washington Wizards with the sixth overall pick in the 2011 NBA Draft. Veselý posted his first double digit scoring night in a 98-77 win over the Detroit Pistons. He finished the night with 10 points, 8 rebounds, and 3 blocks. After Flip Saunders was fired, Veselý saw more minutes on the court under new coach Randy Wittman, who also said that he wants him to shoot the ball a bit more. On April 9, 2012, he recorded his first double-double, finishing the game with 11 points and 11 rebounds in a win over the Bobcats. Veselý continued his success over the Bobcats on April 23, 2012 when he posted a career high 16 points on 8-8 shooting, pulled down 6 rebounds, and recorded 4 steals. He averaged 4.7 points and 4.4 rebounds in his first NBA season.