St John scored 118 goals in 425 appearances - including the winner in the 1965 FA Cup final - after being signed by Bill Shankly from Motherwell in 1961.
Dalglish told Liverpoolfc.com: "I think it's right to use the word 'icon' and I send my prayers to Betsy, his wife, and the family and I just hope they are coping as well as they possibly can.
"He certainly was, and deservedly so, an iconic figure in and around Liverpool Football Club because he came in at the very beginning with Shanks and big Yatesey [Ron Yeats].
"He got off to a terrible start by scoring a hat-trick against Everton at Goodison in the Liverpool Senior Cup, so I think after he did that he was always going to be revered in and around Liverpool.
"It's because of what he did in those early days, Shanks especially, the Saint and Big Yatesey, his big mucker, and it's because of them where this football club is at this particular moment in time. We should be eternally grateful for what he did for the football club.
"I'm sure the family have their own special memories about him as well about their private life, so it is a sad day. Eighty-two is a fantastic innings but it's still sad.
"I'm sure he knew that when he was here, the club really appreciated and revered him for what he did for us."
St John formed a lethal strike partnership with fellow Reds great Roger Hunt.
Dalglish added: "That wasn't a bad partnership was it, Saint and Roger Hunt!
"He scored a great header in extra-time [of the 1965 final] and the FA Cup goes alongside the fact he got two league championship medals after getting promotion from the Second Division with a record number of points.
"So they made people stand up and take notice of Liverpool and because of that the club has moved onwards and upwards. But it wouldn't have gone anywhere without the contribution of people like the Saint."
On his personal memories of St John, Dalglish continued: "He was a funny guy who loved a chuckle!
"The Saint had a lot of friends who will all have a lot of happy memories about him. Although it's a sad day we will remember the happy memories and fun being with him.
"I will always remember him chuckling. I remember him and big Yatesey. When they retired from football they could have been a comedy act on the television, like Morecambe and Wise.
"They were really funny together and the Saint chuckled along."