Dober compiled an outstanding amateur record of 9-0 before turning professional in 2009.
Dober faced Nick Nolte at Bellator 16 on April 29, 2010. He won the fight via arm-triangle choke submission in the first round.
In February 2012, it was revealed that Dober would be a participant on The Ultimate Fighter 15. He was defeated in the entry round by Daron Cruickshank via unanimous decision.
Dober stepped in for an injured Sérgio Moraes to face Sean Spencer at The Ultimate Fighter 18 Finale on November 30, 2013. He lost the fight via unanimous decision.
Dober next faced Nick Hein at UFC Fight Night: Munoz vs. Mousasi on May 31, 2014. He lost the fight via unanimous decision.
Dober then faced Efrain Escudero at UFC 188 on June 13, 2015. He lost the fight by guillotine choke submission in the first round.
Dober next faced Olivier Aubin-Mercier on December 10, 2016 at UFC 206. He lost the fight via submission in the second round.
Dober then faced Josh Burkman on July 29, 2017 at UFC 214. He won by first round knockout.
Drew faced Frank Camacho on January 27, 2018 at UFC on Fox: Jacaré vs. Brunson 2. He won the fight via unanimous decision. This fight earned him the Fight of the Night bonus.