"It is with such sadness that I heard of the passing of one of the true greats of cinema," Craig said in a statement.
"Sir Sean Connery will be remembered as Bond and so much more. He defined an era and a style," he said. "The wit and charm he portrayed on screen could be measured in mega watts; he helped create the modern blockbuster. He will continue to influence actors and film-makers alike for years to come. My thoughts are with his family and loved ones. Wherever he is, I hope there is a golf course."
Craig has played the British secret agent since 2006's Casino Royale. No Time to Die, Craig's final outing as Bond, is set to open in 2021.
Connery played Bond in 1962's Dr. No and five other films. His last appearance as the character was in 1971's Diamonds Are Forever.