With Scott McDonald recovering from hernia surgery and Landry NGuemo on African Nations Cup duty, Mowbray handed a debut to South Korean midfielder Ki Sung-Yeung and gave Darren O'Dea his first game since October following his return from a loan spell at Reading.
But O'Dea was brushed off the ball to allow Carl Finnigan to open the scoring in the 19th minute and Mowbray was forced to hand 18-year-old Josh Thompson a debut when Stephen McManus limped off with a calf problem on the half-hour mark.
Georgios Samaras levelled with a well-taken goal in the 40th minute but Celtic struggled to impose themselves until the final quarter, when they were unable to test Robert Olejnik despite their dominance.
While Wigan signing Caldwell has already been replaced by Jos Hooiveld, the former AIK Stockholm centre-back was not fit enough to come in immediately.
And Celtic missed the influential presence of Robson, who joined Middlesbrough this week along with Willo Flood and Chris Killen.
But Mowbray said: "Decisions had to be made this week. They were made and I have no regrets.
"We can't foresee an illness overnight or injury in training on Friday. That's part and parcel of the job.
"There are no excuses on injuries. I thought the team gave a good account of themselves and yet we didn't get the result."
Mowbray added: "My overriding thought is disappointment we didn't win the game but credit to Falkirk for defending so well and putting their bodies on the line.
"I couldn't question their desire and effort to get a result."
The Celtic manager cannot guarantee reinforcements will arrive quickly and he denied reports he was interested in Portsmouth striker John Utaka.
But he claimed there was no need for major surgery despite falling nine points behind Rangers, who ground out a 1-0 win over Hamilton.
"I see the quality the team possesses," the former West Brom boss said. "We are not far away from being a team that can win consistently.
"Whether we're playing a team at the top of the league like we did two weeks ago or bottom of the league, I don't see much difference.
"I see a team dominating a football match but at this moment not being able to kill teams off.
"I continue to go along the process of building a team and adding the ingredients that are required to do that. The transfer window presents an opportunity to add to this squad.
"We are working hard to do that, to give us the competition to provide the extra edge to turn tight games into victories, but there are no guarantees you can get the players in you are looking for."
Falkirk boss Eddie May was delighted with the draw, which took his team to within a point of Kilmarnock.
However, he was grateful referee Alan Muir did not punish Brian McLean in the closing stages when Marc-Antoine Fortune went down in the box.
"I thought we played some decent stuff in spells and Celtic played extremely well and caused us a lot of problems," May said. "A draw was probably a fair result although I think they should have had a penalty.
"I thought Brian just caught him when he went round him. I don't think he meant it but I've seen them given here."