Morrison - who plays Will Schuester in the hit TV show - was playing a show at New York's 54 Below, with a source telling US Weekly: "Matt came out before the show started and said, 'We lost a great young man too soon'.
"He said Cory was a beautiful person. Then he told the crowd, 'You came out to see a show, I came out to do one, so let's honour Cory with this performance'."
The performer then apparently dedicated a version of A Chorus Line song What I Did For Love to Monteith, which was sung by Michele during the season two premiere of Glee.
The insider concluded: "He was very emotional on stage. It was difficult for him to get up to perform, but he did not want to disappoint anyone."
Monteith's cause of death is currently unknown, with E! Online reporting that an autopsy is due to take place today, Monday July 15, after being fast-tracked in response to "intense public interest".
A Vancouver Police Department source added that the results from the autopsy, including the reasons for his death, will be released in "two or three days".