Sinclair's penalty and a first-half effort by Callum McGregor set them up with a May 27 date with Aberdeen — who beat holders Hibs 3-2 on Saturday — and a potential domestic treble for the first time in 16 years.
McGregor and Sinclair's goals avenged defeat at the hands of Rangers at the same stage last year and extended their unbeaten run to 41 domestic matches this season.
It brings to an end a 92-year Hampden Park hoodoo for Celtic, who failed to beat Rangers in the last four of the competition on their previous five attempts since their last victory at this stage in 1925.
Celtic manager Brendan Rodgers said his side were a joy to watch.
"Everything you would want from your team to play in terms of composure, temperament and technical quality," purred Rodgers. "We restricted Rangers to very few opportunities and the only disappointment was we didn't score more."
His Rangers counterpart Pedro Caixinha felt his side showed Celtic too much respect in the first half.
"At half-time we tried to change some things but I think we needed to be more aggressive in defending," he said. "I think they deserved to win but we could have done, and should have done, much better."
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