Alcaraz scribbled “Valencia” on a television monitor after his victory to show his support for his compatriots who had been hit recent floods.
The French Open and Wimbledon winner was beside himself when he lost his first game in Turin on Monday to the Norwegian Casper Ruud.
And while he used a chest massage during breaks in play on Wednesday, his movements were much sharper.
He came back from 40-15 down to take a 4-3 lead in the opening set after Rublev delivered a crisp forehand at deuce.
And the Russian squandered a 30-0 lead on his next service game as his opponent secured the set.
The eighth-seeded Rublev, who lost his opening game to Germany's Alexander Zverev, held serve and forced a tiebreak in the second round.
An early break gave Alcaraz the edge, although he was pushed back and had to save two set points before securing victory.
Rublev has now lost six straight matches at the ATP finals in straight sets since reaching the semifinals in 2022.
Alcaraz will face Zverev in their final game of the round-robin group stage on Friday.
The German runner-up will face Ruud on Wednesday (19:30 GMT), who will move into the last four in first place and eliminate Rublev if he triumphs.
