German world number three Zverev found himself in deep trouble when trailing by two sets to one against Lajovic, but the Serbian ran out of gas as it finished 2-6, 7-5, 4-6, 6-1, 6-2.
The 21-year-old, a winner of three Masters titles, is yet to reach a Grand Slam quarter-final and was knocked out in the first round at Roland Garros last year.
But Zverev moved into round three for the second time, where he will play Bosnian 26th seed Damir Dzumhur, as he looks to become the first German man to win the title since 1937.
Djokovic advances
Former champion Novak Djokovic reached the French Open third round for the 13th consecutive year by beating Spanish qualifier Jaume Munar 7-6 (7/1), 6-4, 6-4 on Wednesday.
Djokovic, the 2016 Roland Garros champion and former world number one, is seeded 20 this year after struggling to rediscover his best form after returning from a right elbow injury.
He wasn’t at his best on Court Suzanne Lenglen, but goes on to face Spain’s Roberto Bautista Agut for a place in the last 16.
The Serbian 12-time Grand Slam champion fired 39 winners but 33 unforced errors and was broken three times by Munar, the world 155.
“I am very happy to be here on Suzanne Lenglen and see so many people and see the support,” said Djokovic.
“Jaume is a great player, he has lots of quality and good concentration. I congratulate him on a great match today.”
Nishikori survives five-set test
Japan’s Kei Nishikori also staged a comeback to edge out Frenchman Benoit Paire on Philippe Chatrier court, winning 6-3, 2-6, 4-6, 6-2, 6-3 in his first Grand Slam event since Wimbledon last year after recovering from a wrist injury.
Agut through
Bautista Agut, the 13th seed, saw off Colombian qualifier Santiago Giraldo 6-4, 7-5, 6-3, but Czech 17th seed Tomas Berdych lost a rain-interrupted first-round match to home hope Jeremy Chardy on the showpiece Court Philippe Chatrier 7-6 (7/5), 7-6 (10/8), 1-6, 5-7, 6-2.
The unlikely adventure of Argentinian world number 190 Marco Trungelliti, who became the eighth lucky loser in the draw after driving back to Paris from Barcelona, came to an end as he lost 6-1, 7-6 (7/1), 6-1 to Italian Marco Cecchinato in the second round.
Results
1st round
Pierre-Hugues Herbert (FRA) bt Peter Polansky (CAN) 6-3, 4-6, 6-2, 6-2
2nd round
David Goffin (BEL x8) bt Corentin Moutet (FRA) 7-5, 6-0, 6-1
Marco Cecchinato (ITA) bt Marco Trungelliti (ARG) 6-1, 7-6 (7/1), 6-1
Pablo Carreño-Busta (ESP x10) bt Federico Del Bonis (ARG) 7-6 (7/0), 7-6 (7/2), 3-6, 6-4
Roberto Bautista Agut (ESP x13) bt Santiago Giraldo (COL) 6-4, 7-5, 6-3
Novak Djokovic (SRB x20) bt Jaume Munar (ESP) 7-6 (7/1), 6-4, 6-4
Fernando Verdasco (ESP x30) bt Guido Andreozzi (ARG) 6-3, 6-2, 6-2
Damir Džumhur (BIH x26) bt Radu Albot (MDA) 6-3, 6-3, 5-7, 1-6, 7-5
Alexander Zverev (GER x2) bt Dusan Lajovic (SRB) 2-6, 7-5, 4-6, 6-1, 6-2
With inputs from AFP
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