The German-born performer reportedly died at a care home on November 23.
Cleese, 77, the co-creator of the 1970s sitcom, paid tribute to the “gentle and kind” actor, who he said perfectly portrayed the hapless Manuel.
“A very sweet gentle and kind man and a truly great farceur. I first saw him in Habeas Corpus on stage in 1973. I could not have found a better Manuel. Inspired.”
Sachs had been a resident at Denville Hall a private care home in Northwood, west London. Staff said on Thursday night they were unable to talk about his death.
“He had dementia for four years and it wasn’t very pleasant. We didn’t really notice it at first until the memory started going.
After Fawlty Towers, Mr Sachs would go on to play Ramsay Clegg in Coronation Street in 2009 – a year after the Sachsgate scandal in which Jonathan Ross and Russell Brand made a prank call to the actor on the radio about his granddaughter. But he slipped from public life as his illness took hold.
He said on Twitter: “So sad that Andrew Sachs has died. A true friend and a kindred spirit.
Samuel West, whose mother Prunella Scales starred alongside Sachs in Fawlty Towers, added: “Creator of one of our most beloved EU migrants. Such warmth and wit; impossible to think of him without smiling.”
Comedy writer Edgar Wright said Sachs “spun comic gold as Manuel in Fawlty Towers”.