Benik Afone netted twice to put Stoke 2-0 up, before Portuguese forward Marco Matias cut the deficit by half-time.
And with eight minutes left on the clock, Scotland international Bannan stepped up to fire home from 20 yards in front of the Kop.
Owls boss Joss Luhukay made one change, Steven Fletcher coming in for Atdhe Nuhiu up front. New loan signings Michael Hector and Josh Onomah were named on the bench.
But the hosts made a terrible start, Stoke scoring on their first attack in the second minute. The ball found its way to former Owls loanee Afobe, who drilled the ball beyond Cameron Dawson.
Tom Ince then fired just wide as the visitors looked to make it 2-0.
Wednesday struggled to make an early impression, sluggish in possession, and vulnerable in a back four featuring rookies Jordan Thorniley and Matt Penney.
Lucas Joao showed trickery on the ball, but failed to find the target.
And it was no surprise to see Stoke double their lead on 22 minutes. Afobe was allowed to nip in - although there was a hint of offside - and poke the ball home.
But Matias reduced the deficit, almost immediately. He fired home after Barry Bannan's long pass was chested down by Fletcher, to finally lift the home crowd from their seats.
Both teams looked vulnerable at the back, and when Joey Pelupessy robbed Ryan Woods of possession, Adam Reach found himself in acres of space but hammered his shot straight at Jack Butland.
At the other end, Dawson stood strong to block Peter Etebo's close-range effort.
Left-back Penney - who improved as the afternoon went on, alongside Thorniley - almost produced a stunning goal, racing the full length of the pitch only to be denied by England goalkeeper Butland at full stretch.
And the Owls got the equaliser their second-half deserved, when Bannan smashed home a free-kick when left Butland rooted to the spot.
Luhukay said: “It was a very tough game for us. The first 20-25 minutes were very difficult.
“We had too much respect. We must have the confidence in each other. We must believe we can make surprises. We know we must do a lot of things better.
“After the second goal, our fans stayed completely behind the team. At the end we were happy it was 2-2 against a very good team.
“It was a fantastic free-kick from Barry - which was a very important goal for us - and what a pass for the first goal. He gives the team possibilities. If we didn’t have Barry today, it would have been very difficult.
“The character and the personality is 100 per cent. When you have the character and the personality, you can come back.
“The team must have the feeling. When you don’t have the feeling, you must have the mentality.
“You must be happy when you come back from 2-0 down to draw 2-2.”