He had suffered from Alzheimer's disease for several years and died on Monday night, they said.
"Colin will be desperately missed by his family and friends," his family said in a statement.
"Alzheimer's is a cruel illness and there have been difficult times but in the end Colin died peacefully in his sleep.
"We are proud of Colin's many achievements during his life."
The fact-based "Chariots of Fire" revolved around two athletes at the 1924 Olympics: an English Jew who runs to overcome prejudice, and a devout Scottish Christian who runs for the glory of God.
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As an actor, Welland appeared in the popular British television police drama series "Z-Cars" in the 1960s.
He also played a schoolteacher in Ken Loach's heartwarming 1969 film "Kes", about a bullied boy who befriends a kestrel.
Welland is survived by his wife Patricia, four children and six grandchildren.