The Birmingham-born actor, who was the last remaining main cast member from the wartime-set BBC show, died on Friday morning, his agent confirmed.
Hilary Gagan told the PA news agency Lavender had been ill for some time and that his wife and sons were by his side.
He is best known for playing the hapless Pike, who frequently had run-ins with Captain George Mainwaring (Arthur Lowe), calling him "Stupid Boy", and who was looked after by Sergeant Arthur Wilson (John Le Mesurier).
Pike, the youngest member of the Home Guard troop and a bank clerk, would often refer to Wilson as Uncle Arthur as he was in a relationship with his over protective mother Mavis Pike (Janet Davies).
The much-loved Dad's Army series ran from 1968 to 1977.
Lavender acted in other TV comedies such as Yes Minister, Keeping Up Appearances and Goodnight Sweetheart and in a Carry On film.
He also appeared on stage, including in The Merchant Of Venice with Dustin Hoffman, and as the narrator in a touring production of The Rocky Horror Show.
Lavender also played Derek Harkinson, a gay friend of Pauline Fowler (Wendy Richard) in BBC One soap EastEnders from 2001 to 2005.
In the 1970s, Dad's Army regularly attracted more than 18 million viewers, becoming one of the most-watched television programs of its time.
Chris Daniels, director of the annual Bristol comedy film festival Slapstick - which Lavender supported, said the actor "gave our audiences some truly unforgettable moments."
In a statement, he said: "Everyone involved with Slapstick has been deeply saddened to learn of the death of Ian Lavender - a long-standing patron of the festival, regular visitor and valued contributor of programming ideas."
BBC director of comedy Jon Petrie said: "Ian was a much-loved actor and will be sorely missed by all those who knew him.
"In his role of Private Pike, in Dad's Army, he delivered some of the most iconic and loved moments in the history of British comedy. Our thoughts are with his family."
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