Born and raised in Chișinău, Cutelaba began training and competing in Greco-Roman wrestling as an adolescent. He transitioned to mixed martial arts full-time in 2011, and achieved a 1st place finish at the European combat-sambo championships in 2012 for Moldova, in the 90kg category. He is also a former national champion in combat-sambo and judo.

Cuțelaba made his professional debut in April 2012, competing primarily for regional promotions across Eastern Europe. He amassed a record of 11-1 (1) before signing with the UFC in the spring of 2016.

Cuțelaba made his promotional debut against Misha Cirkunov on June 18, 2016 at UFC Fight Night 89. He lost the fight via submission in Cuțelaba next faced Jared Cannonier on December 3, 2016 at The Ultimate Fighter 24 Finale. He lost the fight via unanimous decision. Subsequently, both fighters earned Fight of the Night honors.

Cuțelaba faced Henrique da Silva on June 11, 2017 at UFC Fight Night 110. He won the fight via knockout in the opening minute of the first round.

Cuțelaba was expected to face Gadzhimurad Antigulov on November 4, 2017 at UFC 217. However, Antigulov pulled out of the fight on September 26 citing an injury and was replaced by newcomer Michal Oleksiejczuk. In turn after the weigh-ins, Cuțelaba was pulled from the event by USADA due to a potential Anti-Doping Policy violation stemming from its investigation into voluntary disclosures by Cuțelaba during an out-of-competition sample collections on October 18 and October 19. He was provisionally suspended and the bout was scrapped. USADA handed down a six-month suspension for ozone therapy involving a blood transfusion which is prohibited under 2018 World Anti-Doping Agency (WADA) list on manipulation of blood component. Cuțelaba became eligible to fight again on May 6, 2018.

The bout with Antigulov eventually took place on July 28, 2018 at UFC on Fox 30. Cuțelaba won the fight via TKO in the first round.

Cuțelaba was scheduled to face Glover Teixeira on January 19, 2019 at UFC on ESPN+ 1. However, on January 10, 2019, Cutelaba was pulled out of the bout due to an injury. The pair was rescheduled to meet at UFC Fight Night: Jacaré vs. Hermansson on April 27, 2019. Cuțelaba lost via a rear-naked-choke submission at 3:37 in the second round.