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Alcaraz scribbled ‘Valencia’ on a TV monitor after his victory to show support for compatriots hit by recent floods.
The French Open and Wimbledon champion was out of sorts when losing his first match in Turin on Monday to Norwegian Casper Ruud.
And while he used a chest rub during breaks in play on Wednesday, his movement was much sharper.
He came from 40-15 down to break for a 4-3 lead in the opening set after Rublev fluffed a straightforward forehand at deuce.
And the Russian squandered a 30-0 advantage on his next service game as his opponent secured the set.
Eighth seed Rublev, who lost his opening match to German Alexander Zverev, held serve to force a tie-break in the second.
An early break gave Alcaraz the edge, although he was pegged back and had to save two set points before clinching victory.
Rublev has now lost six successive matches at the ATP Finals in straight sets since reaching the semi-finals in 2022.
Alcaraz will face Zverev on Friday in his final match of the round-robin group stage.
The German second seed is in action on Wednesday (19:30 GMT) against Ruud, who will be the first man into the last four and eliminate Rublev if he triumphs.