A Stephen McManus mistake after five minutes when he was caught in possession at the edge of his own box by Fernandinho led to Brandao's opener.
Three minutes later a stunned Celtic side found themselves two down when Cristiano Lucarelli headed past Artur Boruc.
Brandao had a third goal ruled out for offside and without the heroics of Boruc it could have been a romp for the Ukrainians.
The Hoops are still looking for their first away win in the group stages of the tournament after 13 attempts and look forward to AC Milan visiting Parkhead in their next Group D match.
Strachan said: "We came a long way to play and prepare and after eight minutes we are climbing a mountain which we never expected.
"I didn't expect that to happen as easily as it did, the first goal especially.
"If it is a bit of brilliance you hold your hands up but to give away a goal like that?
"The players were in shock after that but after they recovered they rolled up their sleeves and said 'you will score again over my dead body.'
"I thought the game was running away from us at one point.
"I came with two strikers which might have been naive but I thought we could have got a win.
"We are disappointed for the first 20 minutes but we had chances and if we had scored one I think we could have got back in to the game."
Shakhtar coach Mircea Lucescu insists his side should have hit Celtic for six.
The Romanian, who guided the Ukrainian club to a 3-0 win over the Scots three years ago, believed his side should have scored more than the two goals they did.
Asked what at a fair reflection of the game would have been in terms of the score, Lucescu said: "6-2. A 2-0 win didn't show the game, we had much more opportunities than Celtic.
"I emphasised to my players that we had to have a moment at the beginning of the game, a corner or something, to get Celtic defending and then our technique would take over.
"I congratulate my team because three points are very important.
"If you want to qualify through to the group stages you have to win your home games.
"I was impressed not just by the scoreline but by the way we played."