The actor died on Thursday at a New York hospital following a short illness, with his wife and son at his bedside.
Breaking Bad star Bryan Cranston led the tributes, writing on Instagram: "I am very saddened today to learn of a friend's passing.
"Mark Margolis was a really good actor and a lovely human being. Fun and engaging off the set, and (in the case of Breaking Bad and Your Honor) intimidating and frightening on set."
Bob Odenkirk, the star of spin-off series Better Call Saul, called Margolis a "powerful screen presence".
An official Breaking Bad social media account praised "the immensely talented Mark Margolis, who – with his eyes, a bell, and very few words – turned Hector Salamanca into one of the most unforgettable characters in the history of television".
In the show, Margolis's character – the patriarch of the drug-smuggling Salamanca family, who suffers a stroke after being poisoned by a rival – is only able to communicate by tapping a bell with his finger.
Born in Philadelphia in 1939, Margolis moved to New York to pursue acting.
He forged a successful career as a character actor, with supporting roles in films such as Scarface, Ace Ventura: Pet Detective and Black Swan, as well as HBO series Oz.
In 2012, he was nominated for an Emmy for Breaking Bad.
He is survived by Jacqueline, his wife of 61 years, their only child Morgan and their three grandsons.