Murray needed more than three hours to see off 18-year-old talent Alcaraz, who announced himself on the big stage with a quarter-final run at the US Open.
An Indian Wells runner-up in 2009, Murray was joined in the next round by second seed Tsitsipas and third seed Zverev on Monday (AEDT).
Injuries have struck down Murray in recent years, but the three-time grand slam champion showed there is still plenty of fight left in the tank after rallying past debutant Alcaraz 5-7 6-3 6-2.
Facing a player 16 years his junior, Murray – making his 13th Indian Wells appearance – reached the third round of an ATP Masters 1000 event for the first time since 2016.
The 34-year-old Murray, who hit an underarm ace, improved his record to 27-12 in the desert following three hours, four minutes on court.
"He's obviously got so much potential, so much firepower and these conditions it's not easy to finish points off quickly, but he's able to because he has so much pace from the back of the court so I had to fight extremely hard, coming back from a set down," said Murray, who will next meet Zverev.
"I felt like in the second set he played maybe better. First set I felt like I had more of the opportunities but didn't get it so yeah, happy with the way I fought. He's a top-drawer young player."
Meanwhile, US Open finalist and Olympic gold medallist Alexander Zverev outlasted talented American Jenson Brooksby 6-4 3-6 6-1 to set up a showdown with Murray.
World number four Zverev ended the contest with 12 aces and 28 winners, having qualified for next month's ATP Finals in Turin thanks to the German's four tour-level titles in 2021.
"It wasn't an easy match, but I'm happy to be through, I'm happy to be in the third round and playing Andy now," said Zverev, who has won 18 of his last 19 matches since Wimbledon. "I think he's the only one of the 'Big Four' [including Novak Djokovic, Rafael Nadal and Roger Federer] I haven't beaten yet, so I hope I can change that. I think it's incredible how well he's moving and incredible how well he's playing. I think he's very motivated so I hope I can show my best tennis."
It was far more routine for Greek star Tsitsipas, who eased past Pedro Martinez 6-2 6-4 in his tournament opener.
Tsitsipas needed just 93 minutes to take down his opponent for his Tour-leading 52nd win of the season as the French Open runner-up awaits 25th seed Fabio Fognini for a place in the fourth round.
Italian fifth seed Matteo Berrettini won through to the third round via a 6-4 7-5 success against qualifier Alejandro Tabilo – his first Indian Wells win following two previous appearances.
Felix Auger-Aliassime was the biggest name to depart the event on Monday, with the seventh seed and Flushing Meadows semi-finalist going down 6-4 6-2 to Albert Ramos-Vinolas.
Jannik Sinner, Pablo Carreno Busta, Cristian Garin and Gael Monfils were among the seeds to progress.