[2] After playing youth football for CSKA Sofia and AC Omonia, Galabinov moved to Italy with his parents at the age of 16.

He had loan spells at Sorrento, Bassano, Gubbio and Avellino before leaving Livorno to join Novara in 2015.

Andrey's maternal grandfather Yordan Filipov formerly played as a goalkeeper for CSKA Sofia and the Bulgarian national team.

On 15 July 2013, newly promoted Serie B side Avellino signed Galabinov on a season-long loan deal.

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.

In summer 2017, Galabinov joined Serie A side Genoa on a free transfer after his contract with Novara had expired.

[12] 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.

In his second season at Spezia he earned promotion to the Serie A for the first time in club's history, scoring 7 goals in 19 league games.

On 27 September 2020, he scored Spezia's first-ever Serie A goal, in a 4–1 loss against Sassuolo in the 2nd round of the 2020–21 season at the Stadio Dino Manuzzi.

[17] He made his debut in the match on 5 March, playing in the first 45 minutes and hitting the woodwork after an individual effort.