The competition was run by the University of London Union (ULU) and hosted at a weekly karaoke night, which is put on by an external organisation.
Michael Chessum, president of the ULU, told The Huffington Post UK the incident was being investigated but it was still unclear who picked the controversial winner.
"I'm utterly speechless at this," he said. "Clearly what happened was outrageous, and we'll be investigating it fully."
The ULU is Europe's largest student body, and is made up of several London institutions. Colin Cortbus, a student at London Metropolitan University, one of ULU's members, told the London Student: "ULU will have serious questions to answer over why crowds at its Halloween event cheered on this sick racist dress and even nominated the costume as the best male costume of the night."
Speaking to HuffPost UK, Cortbus added: "At most student unions a racist costume like this one would get you rightly kicked out, at the far-left ULU it gets you an award and a bottle of alcohol. They should be ashamed."
Maham Hashmi-Khan, ULU's black students' officer, added: "How is that possible? They wouldn’t have allowed someone in Nazi costume in. What kind of institution are we running?"