Mowbray was dealt a further blow when defender Glenn Loovens, who had conceded a penalty which allowed Liam Craig to give the home side the interval lead, was carried off with a head injury after clashing with Kenny Deuchar.
It threatened to turn in to another afternoon of frustration for Celtic but the game turned in the visitors' favour after Graham Gartland was sent off seconds in to the second half for a professional foul on Marc-Antoine Fortune.
The former Nancy striker equalised with a wonderful angled-drive in the 64th minute and notched a second from close range, either side of a Georgios Samaras strike, with substitute Paddy McCourt sealing the win with a late solo goal.
Mowbray claims he will start to see his squad replenished ahead of Wednesday's game against Hibs at Parkhead.
He said: "It's just three points for us but the target and challenge for us was to keep chipping away.
"We have another big game on Wednesday against Hibs, a side on form, in confident mood and we are looking forward to that.
"The strength of the squad is going to be significant and the pleasing thing for me is that Scott Brown and Shaun Maloney will probably join in full training tomorrow.
"Scott McDonald is probably Tuesday or Wednesday and Stephen McManus and Mark Wilson are also ready to join in full training.
"So we will be stronger as we move on.
"Everybody is different but Scott Brown will be desperate to put a shirt on for Wednesday after being out for 10 weeks.
"That's the type of boy he is and I will have to assess that, in case he gets injured if he goes in too soon."
Mowbray continued: "Loovens has six stitches in a wound and slight concussion and we will wait and see how he reacts.
"If it turns out to be bad concussion we will obviously not risk him.
"Jos Hooiveld (Celtic's new signing from AIK) has had eight weeks sitting on his backside and we will have to decide, if Loovens doesn't make it, whether he is ready to play or not."
Saints boss Derek McInnes claimed that Gartland's dismissal was the turning point in the game.
He said: "There's no doubt in my mind that we lost the game because we went down to 10 men.
"Players from their team, who you never really saw in the first half, started to get space to work in and they could smell blood.
"They were on the ball and wanted to affect the game.
"But I would suggest we were the better team with 11 men on the park.
"But when you have to stem the tide for 45 minutes it's difficult.
"Celtic are the one team you don't want to go down to 10 men against because they move the ball and isolate you.
"We were vulnerable to their pace and credit to them, there were a couple of quality goals in to the top corner.
"The last 20 minutes was painful but there's not a lot you can do about it."