Law and Rangers' other seven signings cannot complete permanent deals until the club's transfer embargo is lifted in September, but competition rules allowed McCoist to field two as trialists.
Law and Jon Daly got the nod, and the former Motherwell man responded with his side's first two goals in a 4-0 first-round win.
"As trialists go, I thought the two trialists did very well!" McCoist joked afterwards.
"I'm just a little bit disappointed I couldn't get any more of my trialists on, but hopefully we will see them in the near future."
Law was an instant hit with the travelling supporters, and the manager said: "I would consider him to be a big favourite of mine if he keeps that going, never mind the fans.
"His first goal was a great goal because he must have ran 40, 50 yards and you could see a real desire to get on the end of the cross. The second goal as well was absolutely excellent."
Ian Black and David Templeton completed the scoring, and McCoist added: "I thought it was a really polished and comfortable overall performance.
"The standard of goal was absolutely out of this world. The first two goals from Nicky Law were terrific and definitely settled us and allowed us to play the game at our tempo.
"The icing on the cake was Black's finish and Templeton's goal at the end. It's a clean-sheet as well so it's been a good day at the office."