The visitors' early dominance paid off as Shimon Peers grabbed his first goal for the club following a corner.
Moments later he doubled his tally to make it 2-0.
Birmingham responded quickly and halved the deficit soon after but Sheffield made it 3-1 when Goncalo Andrade expertly slotted home after some fine build-up play.
Birmingham came out for the second half on the front foot and before Sheffield knew it they were somehow pegged back to 3-3.
But that is when the visitors stepped it up and went on to score four times without response.
First, Thomas Msadala stole the ball and teed up Adam Derradji.
Seconds later Sheffield’s two-goal advantage was restored as Danny Neath rifled home.
Adam Hollingsworth and Neath notched again late on to add gloss as Sheffield.
Their next game is on Sunday, December 15 when they host Manchester at the EIS.
Head Coach Adam Oldham reflected on the Birmingham win, saying: "It was a hard-fought victory for us.
"We started well and played some exciting futsal at times. I think we learnt a lot as a team and about each other.
"There’s a lot of ability running through this squad."