Scotland are now out of the running for a place in the 2014 World Cup in Brazil and, just as in their 2-1 defeat by Wales on Friday, they lost concentration for five key minutes in the defeat that ended any slim hopes of qualification.
Filip Duricic made the most of their lapses on 60 and 65 minutes to give Serbia three points and leave Scotland bottom of Group A with only two points from six games.
Alan Hutton was caught out by a long ball for the opening goal and was also on the end of a hospital pass from Gary Caldwell for the second, leaving Strachan wondering where his side's concentration had gone.
"When it's 1-0 you have always got a chance of winning but it seems to happen too often that when one goal comes, a second follows," said Strachan, who is still waiting for his first competitive win in charge.
"The same thing happened on Friday so if we lose a goal we really need to see that the game's not finished. There's a time in international football the players have to be told.
"It's a shame for everyone who makes a mistake after putting a lot of good work in. It's not dishonest mistakes.
"It was heavy going and I thought technically they were a right good side. Physically they were stronger than us which gave us problems but they didn't have bigger hearts than us.
"Some of these lads have never played international football before and some of them are playing with injuries but they took the responsibility on their shoulders.
"The longer the game went on the better it got for us. We were happy with loads of things but it happens too often that one goal goes in and another one quickly follows so we have to remedy that.
"It's easy to work on people who are good people. It's not a quick fix to get as strong and as technically gifted as that. But there's lots of things I know I can work on.
"I said that we might have to take a few slaps but we have to do something, we can't keep plodding along and hoping something happens. I want to pick players that will make things happen.
"Unfortunately for players like Shaun Maloney and Steven Naismith, the conditions weren't great for them but they can't be embarrassed. Sometimes when you have played a game of football, as long as you have given it your best shot you can hold your head up high. Sometimes the other team is better and that is what happened."