The comedian and actor said the style of relaxed, informal documentary making, with which he has had success on television, owes much to what he learned from Peat in the 1970s.
"I shall miss him and Scotland shall miss him," the Glaswegian comedian told The Herald.
Peat, with friend and colleague Murray Grigor, worked with Connolly for two early and influential films, beginning with Clydescope, a 1974 film which featured the youthful ex-shipbuilder and budding comedian.
Later, Peat and Grigor made Big Banana Feet, an observational documentary which followed Connolly on tour in 1975 when he played two sell-out gigs at Belfast and Dublin.
Connolly said that Peat's work had inspired him in the way he makes his films, such as his World Tour series of travel documentaries.
He said: "I owe David so much when it comes to the style of reality/documentary style of filming that I have been doing over the past few years.
"The relaxed style of presentation, the readiness to change the subject at the drop of a hat when a likely situation arose, and the lack of TV cliche 'walk and talk' style of doing things, I learned it all from David and Murray in those films which seem like yesterday, although they do go back a wee bit.
"The main thing I learned from David and Murray is that you mustn't lose sight of the fact that it is supposed to be fun, that we are all very lucky to be doing it."
Peat and Connolly recently met again when the Scottish comedian was on tour, with Peat gathering material for a retrospective documentary about his own life in television and film.
He added: "If you can manage to get those feelings to come up the lens then you have got it made.
"David never failed, in my experience, to do just that."
Peat, who was suffering from a type of cancer called myeloma, was working until recently on a BBC Scotland programme about his work as a film-maker and a photographer – A Life in Film: David Peat.
Born in 1947, Peat was one of Scotland's most respected documentary film-makers and cameramen. His list of television credits include Gutted, a BBC Scotland film about the plight of the fishing industry, This Mine Is Ours, about Monktonhall Colliery, and series such as Clydebuilt and Scotland On Film. He became an expert in sensitive, observational camera work.
Connolly added: "I'm glad to have had the privilege of spending time with him, both in the early days, and recently in Blackpool during my tour when he came to interview me as part of his television retrospective of his work.
"It was years since we had met, but we picked up the conversation exactly where we had left it off all those years ago, dead easy.
"I shall miss him, and Scotland shall miss him."