Annan went 2-0 up against the Scottish giants before seeing out a famous victory, handing Rangers their first Third Division home defeat in the process.
McCoist revealed that he didn't criticise the players in the dressing room, but let them know that their performance was not good enough.
"It was a really, really poor 90 minutes. The players know what is expected of them at this club and they dropped below those standards," said McCoist.
"I didn't have harsh words with them in the dressing room, I just told them that it was unacceptable."
Despite the defeat, their second of the season, Rangers still lead the Third Division table by 17 points.