The 20-year-old midfielder was a key part of his side's 3-0 win against Tottenham Hotspur, grabbing the third and final goal.
Morrison, who joined the Hammers from Manchester United, spent last season on loan at Birmingham City and encountered some difficulties with question marks over his attitude.
However, he has forced his way into the West Ham first team this season.
"He's 20, but he's quite old in terms of years but he's learnt so much in a short period of time. That's the beauty. The penny has dropped," Allardyce told BBC Sport.
"While [his former manager] Sir Alex Ferguson told me I hope you can sort him out because you will have a top-class player on the your hands, I'm not sure I've done that but he's enjoying his football. He can do something no-one else can do and we saw that today."
West Ham's win was their first on the road this season.