Villa's last triumph in the competition was in 1957 and McLeish has witnessed enough upsets throughout his career not to take anything for granted when they face League Two side Bristol Rovers at the Memorial Ground on Saturday.
But he can also draw on last season's Carling Cup triumph with Birmingham as a pointer that anything is possible with the aid of favourable draws and a degree of good fortune.
McLeish said: "It (1957) is a long time isn't it? But Hibernian had gone something like 100 years without winning the Scottish Cup when I got them to the final.
"I know how much it hurts the fans when they've not had success for a long, long time - especially a club of Aston Villa's history and heritage.
"Certainly as a kid living in Scotland, I loved tuning into the FA Cup. I remember watching Hereford beating Newcastle, Ronnie Radford's goal, seeing all the brilliant drama.
"I hope there's more drama to come and it's all in Aston Villa's favour in this competition. We seriously want to do well in the FA Cup.
"We saw what happened with Birmingham last season. It can be done, absolutely, but you've got to do your best and sometimes you need a wee rub of the green in terms of draws."
McLeish will not take anything for granted despite Rovers sacking manager Paul Buckle this week and sliding towards the bottom of the table.
He said: "From the outside, it might look like they are in a bit of a trouble after losing their manager.
"But they will see it as a game where they have absolutely nothing to lose.
"They are not expected to beat Aston Villa but they will be asking our players if they fancy it.
"I remember going to little grounds with my own Scottish Cup successes with Aberdeen and you knew you had to roll your sleeves up.
"With the romance of a giant-killing in the FA Cup we have to be at our best to avoid that."
