Massimo Donati's winner had not looked like coming in a tight second half, which followed an eventful first period.
Brandao gave the visitors a fourth-minute opener after a mistake by John Kennedy, who was later carried off on a stretcher after Lee Naylor had suffered the same fate.
Paul Hartley had to move to left-back while Steven Pressley came on for his first game of the season and Brandao was twice denied by Artur Boruc before Jiri Jarosik fired home a superb volley in the 45th minute.
Jarosik missed two great headed opportunities and time looked to have run out before Donati found the bottom corner through a crowded penalty area.
"In different games we have had to think, really play well technically but for sheer guts that's as good as it gets," Strachan said. "Just when you think you have seen it all, you think about every scenario, but you can never know what's going to happen.
"Losing a stupid goal, ending up with a midfielder playing left-back, a centre-half at right-back (Gary Caldwell) and someone who has not played for five months who is 34 (Pressley).
"How did we win the game? The simple answer is with a huge heart."
Strachan revealed Naylor's injury was only a calf strain but Kennedy's problem is causing much more concern.
The defender returned from a three-year absence with a knee injury in March and looked in distress after contesting a high ball.
"John has gone to hospital," Strachan said.
"With John's record we have to be spot-on. We keep our fingers crossed."