Rob Dowd and Colton Fretter are out for weeks and Guillaume Desbiens is not likely to be available for another two weeks.
Tyler Mosienko, Rod Sarich and Levi Nelson are all struggling, but they turned out for the team at Sheffield Arena in a barnstorming 3-2 win which coach Paul Thompson described as "bloody marvellous."
Sheffield took two points over the division's pace-setters with a last period rally, after going behind.
Thompson said he was a "proud coach tonight."
His side is now four points behind Cardiff, but with two games in hand.
Pinc, however, is a grave loss.
The Czech netminder damaged his left shoulder in a save at the start of the second period and had to be substituted by Brad Day.
He was taken to hospital for treatment.
Steelers, in second place, started off the night six points behind the Welsh, who have added two new quality recruits recently in Gleason Fournier and Ryan Russell.
Steelers had plenty of chances in an error-strewn first period, Jason Hewitt and Mathieu Roy both failing to break the deadlock on breakaways.
Likewise league top scorer Joey Martin back-handed a golden Cardiff opportunity wide.
The Welsh took the lead at 19;43, Ryan Russell finding the net.
The second period was a tale of two goalies.
Pinc went off 34 seconds in with his injury but Ben Bowns stonewalled attempt after attempt from the home side.
Levi Nelson looked the player most likely to score, but Bowns seemed to have everything covered, particularly his near post.
Jason Hewitt scored an equaliser to trigger a wage of Sheffield attacks in the third period.
Then Conny Stomberg's shot was deflected by Fredrik Vestberg to give Steelers the lead.
And the game was sealed by a Ryan Hayes empty net goal in the last two minutes, although Carl Hudson scored with 12 seconds left for Devils.
Another roller-coaster night at Sheffield Steelers...
Saturday 23rd January
Belfast Giants 6 Coventry Blaze 3
Braehead Clan 4 Dundee Stars 2
Fife Flyers 7 Edinburgh Capitals 1
Nottingham Panthers 4 Manchester Storm 0
Sheffield Steelers 3 Cardiff Devils 2
Sunday 24th January
Cardiff Devils v Fife Flyers – 6.00pm
Dundee Stars v Sheffield Steelers – 6.30pm
Edinburgh Capitals v Braehead Clan – 5.30pm
Manchester Storm v Nottingham Panthers – 5.00pm