Andrey was born in Sofia to Asen Galabinov and Polina Filipova. His father is a former national volleyball team player and his mother also played the sport. Andrey's grandfather Yordan Filipov formerly played as a goalkeeper for CSKA Sofia and the Bulgarian national football team. His other grandfather was a volleyball player and coach.
Galabinov began his football career with CSKA Sofia. In 2004, he moved to Cyprus with his family and joined AC Omonia youth academy. A year later, his father had been appointed as a coach of Pallavolo Modena and Andrey moved to Italy. In 2006, he signed with Castellarano.
In July 2007, Galabinov joined Bologna but could not break his way into the first team and went out on loan to Giulianova, and then to Giacomense.
On 1 February 2011, Galabinov signed with Livorno on a four-and-a-half-year deal for an undisclosed fee. He made his Serie B debut on 7 February, in a 0–1 home loss against Vicenza. He appeared in only three league games to the end of the season. In an attempt to earn some playing time, Galabinov spent the 2011–12 season on loan in the Serie C1, where he played for Sorrento and Bassano.
In July 2012, Galabinov moved to Gubbio. The move was agreed as a one-year loan deal.
On 15 July 2013, newly promoted Serie B side Avellino signed Galabinov on a season-long loan deal. He made his debut in a 2–1 home win over Novara on 24 August, playing the full 90 minutes. On 30 September, he scored his first goal, netting a 35th-minute penalty in a 1–0 win over Empoli. On 19 October, Galabinov scored a twice in a resounding 4–1 home victory against Carpi. He continued his goal-scoring form with another brace against Cittadella, ten days later. On 25 January 2014, he netted Avelino's only goal in their 2–1 defeat at Novara, taking his tally of league goals this season to 10. Galabinov made 39 appearances during the 2013–14 season finishing as the club's top scorer with 15 goals.
Galabinov returned to Livorno at the end of the season, scoring eight goals in six pre-season friendly games. On 17 August 2014, he scored his first competitive goal for Livorno in a 2–4 Coppa Italia loss against Bassano Virtus. He then continued his goalscoring form by scoring his first league goal for the club on the opening day of the 2014–15 season in Serie B, a 1–1 draw against Carpi at Stadio Armando Picchi on 30 August. Two months later, on 1 November, Galabinov scored a penalty in a 3–2 home win over Bologna. He left Livorno after his contract expired in summer 2015.
On 12 August 2015, Galabinov signed as a free agent with Novara. At first he wore no.35 shirt, but picked no.16 after Nicolas Schiavi switched to no.20.
In summer 2017, Galabinov joined Serie A side Genoa on a free transfer after his contract with Novara had expired. He scored his first goal for the club on 26 August 2017, on the second matchday of the season, converting a penalty for a 2–0 lead in a match that ended in a 2–4 home defeat to Juventus.
On 20 February 2014, Galabinov was called up to the Bulgaria squad, for their friendly fixture against Belarus. He made his debut in the match on 5 March, playing in the first 45 minutes and hitting the woodwork after an individual effort. Galabinov opened his account for the national side on 23 May 2014, scoring the first goal in the 1–1 draw against Canada.As of 8 June 2017
As of match played on 16 November 2014.