Strachan admitted his team faded late in the second half as they sought to get back on level terms after Rickie Lambert's goalscoring introduction.
The visitors had gone in front early on in the international friendly through James Morrison and then again soon after the restart when Kenny Miller got the better of Gary Cahill.
However, equalisers from Theo Walcott and Danny Welbeck pegged them back before Lambert came on to head the winner from a corner.
"We had a problem physically in the last 20 minutes," admitted the Scotland boss.
"We were trying everything to score an equaliser. I tried to keep Shaun (Maloney) on as long as possible.
"If you look at it, when we had 11 men and in free [open] play, England couldn't score. They scored when we were down to 10 men [Grant Hanley was off for treatment when Walcott hit England's first goal] and they scored from set plays, which was hugely disappointing.
"But there are top players in that England side. We have good players. To get great victories, we have to make no mistakes whatsoever."