The Cumbrians came from behind and scored twice in the final five minutes to leave them on the verge of a fourth consecutive win under Kavanagh's stewardship.
However, Shrewsbury winger Paul Parry then found the net deep in stoppage time to earn the hosts a last-gasp point.
Kavanagh, who has openly admitted that he would like to manage Carlisle on a permanent basis, was upbeat despite losing his 100% record.
"It's that never-say-die attitude," he told reporters. "I said to the players that Shrewsbury have conceded late goals. They've drawn six, lost three and won one of their 10 games.
"I felt it was going to be our day, especially when it went to 2-1, but it wasn't to be. The dressing room afterwards was pretty quiet.
"They were euphoric at 2-1 and I just laughed at the end of it and said 'look boys, this is the start of something good hopefully'."
Next up for Carlisle is a trip to Morecambe in the Johnstone's Paint Trophy on Tuesday night.