Starting his footballing career in his native country, Georgia. Qazaishvili made several appearances for Metalurgi Rustavi and Sioni whilst on loan from Saburtalo Tbilisi before joining Dutch side Vitesse in 2011. After impressing with the Vitesse youth sides, Qazaishvili was given his debut on 27 November 2011, in a 0–0 draw with Twente, starting the fixture before being replaced by Georgian teammate Giorgi Chanturia in the 65th minute. A week later, Qazaishvili scored his first Vitesse goal in a 4–0 victory over RKC Waalwijk, netting in the 21st minute to make it 2–0. In the later years of Qazaishvili's spell with Vitesse, he became a key figure, especially in their 2014–15 and 2015–16 campaigns, netting nineteen times between the two.
On 31 August 2016, Qazaishvili joined Polish side Legia Warsaw on a season-long loan. On 14 September 2016, Qazaishvili made his Legia Warsaw debut in a 6–0 home defeat against Borussia Dortmund in the opening matchday of the 2016–17 UEFA Champions League. On 1 October 2016, Qazaishvili made his league debut for Legia Warsaw in a 3–0 victory over Lechia Gdańsk, replacing Guilherme in the 76th minute. On 14 May 2017, Qazaishvili scored his first and only goal for Legia Warsaw in their 6–0 home victory over Nieciecza, coming off the bench to seal the win, netting in the 88th minute.
Following the conclusion of the 2016–17 campaign, Qazaishvili returned to Vitesse after appearing sixteen times and scoring once (in all competitions) for Legia Warsaw. After leaving Legia, he stated that he was disappointed with the loan and with Legia coach Jacek Magiera, who Qazaishvili said told him from the very beginning of the loan that he did not have a place with the team. However, he was pleased to have the opportunity to play in the Champions League and alongside players like Nemanja Nikolić, who he would later compete against in MLS.
On 22 June 2017, Qazaishvili joined American side San Jose Earthquakes as a Designated Player, the team's youngest ever, after a six-year spell with Vitesse. He was introduced at the club on July 10, 2017, by General Manager Jesse Fioranelli during halftime of San Jose's U.S. Open Cup quarterfinal victory over the LA Galaxy. His club debut came in a friendly match four days later on July 14 as a 63rd minute substitution in San Jose's 4-1 defeat of Eintracht Frankfurt. Vako scored in his MLS debut on July 19, coming on as a halftime substitution for Danny Hoesen against the New York Red Bulls at Red Bull Arena and scoring San Jose's lone goal of the 5-1 defeat in the 88th minute off of a cross by Jahmir Hyka. He first started for the Earthquakes on August 9 during the team's U.S. Open Cup semifinal loss against Sporting Kansas City, earning an assist on Hoesen's goal and converting his penalty when the game was settled with a penalty shootout. During his first MLS start, at home against the Philadelphia Union on August 19, Vako scored his second league goal, assisted by Tommy Thompson.
Qazaishvili has represented Georgia at under-17, under-19 and under-21 levels before making his senior debut in 2014 in 2–0 victory over Liechtenstein, in which he replaced Avtandil Ebralidze at half-time.