The Glasgow giants saw a run of 20 straight victories come to an end on Boxing Day when Jamie Longworth handed the Stair Park outfit a shock point with his last-gasp leveller.
McCoist's side took out their frustration at missing out on setting a new club record run of results by putting four past Dunfermline on Monday night.
But Law confessed it is only now that he is beginning to grasp the enormity of the expectations that come with being a Rangers player.
He said: "You are told as soon as you join Rangers that the standards are set high here but you probably don't realise just how high until you suffer a setback - only then you can see for yourself.
"It was like the end of the world after the Stranraer game. The stadium had been full, then it was empty. From about 60 minutes in the fans showed their displeasure with the performance and rightly so.
"They pay their money and expect the best, but we didn't give them that.
"It was an eye-opener for me and a lot of the other new boys. We had won every single game since we had been here. Of course it was never going to last forever but to see the reaction to a draw, not even a defeat, was incredible. I wonder what it will be like if we do lose.
"However, we all hope we have repaid that a little with the result at Dunfermline. To win 4-0 against the side second in the league, a full-time side, was a great response. Now we just want to go on another run like the one we've just been on."
Law struck Rangers' third goal against the Pars after openers from Fraser Aird and Nicky Clark, while Robbie Crawford added a fourth in stoppage time.
That resounding victory came after McCoist demanded an improved showing.
But former Motherwell playmaker Law insists he and his colleagues are not struggling for motivation despite leading League One by 14 points with a game in hand.
"It's not hard to get ourselves up for these games," said the Englishman.
"Looking back at the Boxing Day game, it was maybe a bit of complacency from ourselves that cost us.
"We were poor but should have still seen the game out. To concede with the last kick of the ball was really just a lack of concentration.
"We should never have conceded that goal but we have learned from it and the Dunfermline game was all about how we responded.
"The manager demanded a response and he was delighted with what he got."