Cowan netted 28 goals for the Edinburgh club back in the 1985/86 season but that record is in danger of being beaten with Wolves loanee Griffiths currently sitting on 27 goals with three league fixtures and the Scottish Cup final against Celtic on May 26 remaining.
"I would expect him to beat my tally. He has got the ability, the talent and certainly showed a maturity on the park this year," Cowan told SunSport ahead of Hibs' Edinburgh derby with Heart of Midlothian on Saturday.
"I'm surprised that my record has stood for so long.
"As each year goes on, you think it gets tougher to break it but with Leigh, you've got something that is a bit extra special.
"He goes on to the pitch full of confidence every time and probably expects to score in every game.
"I'd not be surprised if he scored one, or even two, at the weekend - although I think it'll be a tougher game for Hibs than it will be for Hearts.
"I don't think he'll be thinking of the record - he'll just be looking to score goals.
"He has added to his game as a striker.
"In the past year, he has scored spectacular goals but this season I've seen him score tap-ins and ones in and around the six-yard box.
"He has brought a different dimension to his game.
"I've also seen an improvement in his link-up play in terms of bringing the ball in and bringing in other players."