Smith, 60, who suffered poor health for a number of years, was found dead on Friday morning at his north London home, his agent Michael Foster confirmed.
A larger-than-life figure, known for his lugubrious, hangdog expression, Smith's breakthrough came when he joined Rowan Atkinson, Pamela Stephenson and Griff Rhys Jones in the topical series Not The Nine O'Clock News, which became a cult BBC hit in the early 80s.
He then formed a hugely successful double act with Jones, in the sketch series Alas Smith and Jones, a screen partnership which lasted for 10 series over 16 years.
A writer, producer and theatre director, Smith also directed six feature films, including The Tall Guy and Bean, starring Rowan Atkinson as the silent comic character.
The independent production company TalkBack Productions, which Smith and Rhys Jones formed in 1981, also spawned hit television series including I'm Alan Partridge and was sold for £62 million in 2000.
Smith admitted that he suffered from depression and battled an addiction to painkillers, ending up in hospital in 1999 with severe stomach ulcers.
He appeared pale and slurred his words during an appearance on an edition of Celebrity Mastermind and was treated for a virulent throat infection in 2009.
Smith's screen appearances became scarce in recent years. His last role was a hotel manager in Stephen Poliakoff's BBC2 drama series Dancing On The Edge earlier this year.
Griff Rhys Jones led the tributes, describing Smith as "a force for life" to everybody who met him. His comedy partner was "a gentleman and a scholar, a gambler and a wit. We are all in a state of shock. We have lost a very, very dear friend."
Smith attended Oxford University while Jones was at Cambridge and the pair first became known to each other while performing at the Edinburgh fringe. Jones added: "I will look back on the days working with him as some of the funniest times that I have ever spent. We probably enjoyed ourselves far too much, but we had a rollercoaster of a ride along the way. He was a very generous and supportive actor"
John Lloyd, the Not The Nine O'Clock News producer said his friend had been ill for some time. "Mel did an extraordinary thing - he taught us all how to make comedy natural," he told the BBC. "Mel was a very kind person and open to new ideas. He would always take on a part in Not if the others didn't want to do it."
Lloyd added that Smith "liked a drink" and recalled their last lunch together, which began at Claridge's at 12.30pm and ended at 9pm when Gordon Ramsay joined them for coffee.
Stephen Fry tweeted: "Terrible news about my old friend Mel Smith, dead today from a heart attack. Mel lived a full life, but was kind, funny & wonderful to know."
Graham Linehan, the Father Ted co-writer, said he was "very sad" to hear the news. "He and Griff gave Arthur (Mathews, Father Ted co-creator) and I our break. Was always so kind and generous to us."
Chris Langham, an original member of the Not the Nine O'Clock News team who, while preparing for a comedy about a psychiatrist, was convicted of downloading obscene images of children, said last night: "Mel was one of the people who really stood up for me when I was having difficulties. He was a funny guy who did naturally what it took many people a great deal of hard work to achieve. He had a gift for comedy and was also a really kind man. What more do you want? It's a really sad loss."
BBC director general Tony Hall said: "Mel Smith's contribution to British comedy cannot be overstated. On screen he helped to define a new style of comedy from the late 1970s that continues to influence people to this day."
Matt Lucas, of Little Britain fame added on Twitter: "So sad to hear about the passing of the great Mel Smith. Brilliant writer, actor and director and a lovely man too."
Smith was described as having "extraordinary natural talent" by Peter Fincham, director of television at ITV.
Mr Fincham, who became agent to Smith and Jones as well as their business partner at Talkback, said: "He had had extraordinary natural talent with the rare gift of wearing it lightly. Mel and Griff were one of the great comedy acts and it's hard to imagine that one of them is no longer with us."
In a statement released on behalf of Smith's wife Pam, Mr Foster said: "Mel Smith, comedian and writer, died on Friday aged 60, from a heart attack at his home in north west London."
News of the London-born Smith's death was first revealed on Saturday afternoon by the Australian novelist Kathy Lette who tweeted: "R.I.P. Mel Smith. Sorry to bring sad and bad news, but apparently he died from heart attack in his sleep."
'Mel was a force for life': Griff Rhys Jones' tribute to Mel Smith
"I still can't believe this has happened. To everybody who ever met him, Mel was a force for life. He had a relish for it that seemed utterly inexhaustible.
"He inspired love and utter loyalty and he gave it in return. I will look back on the days working with him as some of the funniest times that I have ever spent.
"We probably enjoyed ourselves far too much, but we had a rollercoaster of a ride along the way. Terrific business. Fantastic fun, making shows. Huge parties and crazy times. And Mel was always ready to be supportive. Nobody could have been easier to work with.
"We never had an argument about which part we should play or how we were going to do something. We never had an argument, in fact. We loved performing together. He was a very generous and supportive actor. We had a good deal of fun.
"Mel was not a pressure person. He was a gentleman and a scholar, a gambler and a wit. And he was a brilliant actor. But he never took himself or the business too seriously. We are all in a state of shock. We have lost a very, very dear friend."