He was eight years old when he began training for the Red Star Belgrade, where he passed all age categories. Between 2007 and 2009 he played for FK Rad before returning to Red Star where he would play in the first squad until 2010 when he moved to France and signed with Bordeaux. In January 2012, he went on loan to Dynamo Dresden for the rest of 2011–12 season. On 23 January 2015 Savić signed for English Championship club, Watford until the end of the season. However, he was one of three released by Watford following their promotion to the Barclays Premier League in 2015, and left without making an appearance for the club. In 2015, Savić joined Sheriff Tiraspol, where he made a total of 40 appearances and 4 goals in all competitions over two seasons. He scored a bicycle kick in a league match against FC Zaria Bălți on September 19, 2015.
In June 2017, Savić returned to his home club, Red Star Belgrade on a two-year contract. In his first season back in Belgrade, Red Star became the first team in history to make it to the 2018 Europa League knockout phase from the first qualifying round. It was also Red Star's first season surviving the group stage of a UEFA competition in 26 years. During the 2017 season, coach Vladan Milojević played Savić in reputable defensive partnerships with Srđan Babić and Damien Le Tallec. With Savić, Red Star conceded only two goals in the 2017 Europa League group stage; it was the second best defense in the Europa League group stage behind eventual semi-finalists Red Bull Salzburg. In May 2018, Savić was elected in the best 11 players for the 2017–18 Serbian SuperLiga season, by clubs captains' and managers' choice. The following season, Savić managed to guide Red Star to their first ever UEFA Champions League appearance. During the qualifying rounds, playing mostly in tandem with Miloš Degenek, Red Star managed to concede only three goals. In absence of club captain Nenad Milijaš, Savić captained Red Star in all four of the games he featured in. In two out of those four games, both home against Napoli and Liverpool, Red Star managed to keep a clean sheet. He played the first half against Liverpool at Anfield but was substituted at half-time due to injury. He missed the remaining two games against Napoli in Naples and Paris Saint-Germain in Belgrade also because of injury. On 25 January 2019, Savić extended his contract with Red Star to the summer of 2022.
On 14 July 2019, Cypriot club APOEL officially announced Savić's signing. Savić joined APOEL from Red Star on a free transfer, with a provision that Red Star be paid half of the sum of his next transfer. He signed a three-year contract. In late August 2019, he underwent surgery for a groin injury.
Vujadin is a son of Serbian former professional footballer Dušan Savić. He has three children with unmarried spouse Mirka Vasiljević, who is an actress. Savić was named after Serbian coach and former player Vujadin Boškov. He is also nicknamed Giška after his relative Đorđe Božović.
