Celtic then exerted sustained pressure but visiting goalkeeper Robert Olejnik was only seriously troubled once in the second half, when he produced a superb flying save from debutant Ki Sung Yeung's 30-yard free-kick.
The finale drew comparisons with Celtic's previous home game, when they drew 1-1 against Rangers despite creating the vast majority of chances.
Celtic have now failed to win 12 of their last 19 games and are nine points behind Clydesdale Bank Premier League leaders Rangers, although they have a game in hand against struggling Kilmarnock.
And Mowbray has faith in his players to claw back the deficit.
"The team was good," he said. "We didn't win the football match, like we didn't two weeks ago, when the team was fine again.
"If I thought there was major surgery to do, I would be more concerned.
"I think the players have high enough quality to turn these games into victories.
"Somewhere along the line we are going to score a lot of goals, I'm sure.
"We are in decent shape. You talk about us being nine points behind but let's judge where the season finishes and I've got every confidence in this group of players."
Celtic found themselves a goal behind after Darren O'Dea, making his first appearance since playing on loan for Reading in October, failed to clear and allowed Finnigan to net in the 19th minute.
Artur Boruc then saved brilliantly from Ryan Flynn before skipper Stephen McManus suffered a calf injury and was replaced by 18-year-old Josh Thompson.
Mowbray had already been without six players who featured against Rangers, including the departed Gary Caldwell and Barry Robson and African Nations Cup participant Landry N'Guemo.
While Mowbray was not helped by illness to Glenn Loovens and a late groin injury for Danny Fox, whose absence came on the back of Scott McDonald's hernia surgery, the manager could not guarantee there would be additions to his squad before the transfer window closes.
However, full-back Andreas Hinkel also retains confidence in the current squad and argued that playing without the pressure of leading the table could be an advantage.
"We still have the quality to win the league," the German said.
"It's very disappointing at the moment but it changes so quickly.
"At the end of this month it can be totally different, there are so many points to play for.
"We have to do better but sometimes it's easier to concentrate on your job when you are behind.
"I think we can come back, we still have it in our own hands."
But Hinkel admits that Celtic cannot go on failing to take advantage of their dominance.
"It's happened too often this season that we had to come from behind," the 27-year-old said.
"It's strange, because it's always the same this season. We always control the game and have enough chances and then give silly goals away.
"Then we have to fight back and it's always difficult."
Scottish Premier Winner 2009/10 Win Outright: Celtic 2/1