WIMBLEDON, England—Serena Williams kept telling herself she was facing just another foe in the Wimbledon final Saturday, just another woman who hits the ball quite hard, just another player trying to deny her a Grand Slam title.

She wasn’t facing just anyone, of course. She was playing her older sister Venus. And when the latest all-Williams final finished, when Serena wrapped up a 7-6 (3), 6-2 victory for a third Wimbledon championship and 11th major title overall, she jogged to the net with her arm extended for a handshake.

Venus pulled her close for a warm embrace, i...

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