Fleming and Wimbledon doubles champion Marray took to the court at the Ricoh Arena needing to defeat Kunitsyn and Baluda to keep Britain's hopes of winning the five rubber tie alive after two losses on Friday.
Dan Evans and Jamie Ward lost to Dmitry Tursunov and Evgeny Donskoy respectively in five-set thrillers as Russia took a 2-0 lead however Fleming and Marray prevented them taking the tie.
They saw off Kunitsyn and Baluda 6-1, 6-4, 6-2 in less than an hour and a half to make it 2-1 in the Euro/Africa Zone Group I second round clash with a place in the World Group play-off on offer.
Fleming is playing in his ninth Davis Cup rubber and usually partners Hutchins, who is recovering from Hogdkin's lymphoma but has helped the British team prepare for the tie.
Ward faces Tursunov and Evans Donskoy on the final day of singles matches on Sunday and Fleming devoted his and Marray's win to Hutchins while also issuing a rally cry to his other team-mates.
"I love playing Davis Cup. I want to dedicate this win to Ross, he is a big part of this team," said Fleming. "Let's do it tomorrow boys. We can do this. Let's go."
And Britain captain Leon Smith is daring to dream of wins for Ward and Evans which would secure the team a place in the World Group play-off.
"That was outstanding what they did there; they stamped their authority on to the match from start to finish," said Smith.
"Also the crowd were excellent so thank you very much. Upsets can happen. Give the guys whatever you have got tomorrow please."