World number one Djokovic swept past 153rd-ranked Galan 6-0, 6-3, 6-2, his most challenging moment coming when he helped groundstaff lay more clay on the surface of Court Philippe Chatrier after rain had made it treacherous.
"I played pretty solid today, I'm happy about that," said Djokovic, the 2016 champion who is chasing an 18th Grand Slam title, after his 71st win in Paris.
Djokovic, who has dropped just 15 games in three rounds, took his 2020 record to 34 wins against one loss as he continues his bid to become the first man in half a century to win all four Grand Slam titles twice.
Next up is Russian 15th seed Karen Khachanov who put out Chile's Cristian Garin.
The day after world number 239 Hugo Gaston and Sebastian Korda, ranked at 213, made the last 16, German qualifier Altmaier kept the sport's outsiders in the headlines.
The world number 186 stunned seventh seed Matteo Berrettini of Italy 6-2, 7-6 (7/5), 6-4, becoming just the fourth man in 20 years to reach the round of 16 on his Grand Slam debut.
Victory also means an unexpected financial windfall for the 22-year-old who is guaranteed at least $221 400 having earned $173 600 in his entire six-year career.
Altmaier paid tribute to three-time major winner Stan Wawrinka for his help when he missed virtually all of 2018 through injury.
"When I was watching him, he always says 'Allez, Stan.' I copy him a little bit because I like to say, 'Allez, Dan'. He has been a huge reason why I'm healthy now."
French Open results at Roland Garros on Saturday, the seventh day of the delayed 2020 championship (x denotes seeded player):
Men
3rd rd
Novak Djokovic (SRB x1) bt Daniel Galan (COL) 6-0, 6-3, 6-2
Karen Khachanov (RUS x15) bt Christian Garn (CHI x20) 6-2, 3-6, 6-4, 6-2
Pablo Carreno-Busta (ESP x17) bt Roberto Bautista Agut (ESP x10) 6-4, 6-3, 5-7, 6-4
Daniel Altmaier (GER) bt Matteo Berrettini (ITA x7) 6-2, 7-6 (7/5), 6-4
Marton Fucsovics (HUN) bt Thiago Monteiro (BRA) 7-5, 6-1, 6-3
Andrey Rublev (RUS x13) bt Kevin Anderson (RSA) 6-3, 6-2, 6-3
Grigor Dimitrov (BUL x18) bt Roberto Carballes Baena (ESP) 6-1, 6-3 - retired
Stefanos Tsitsipas (GRE x5) bt Aljaz Bedene (SLO) 6-1, 6-2, 3-1 - retired
Women
3rd rd
Ons Jabeur (TUN x30) bt Aryna Sabalenka (BLR x8) 7-6 (9/7), 2-6, 6-3
Fiona Ferro (FRA) bt Patricia Maria Tig (ROU) 7-6 (9/7), 4-6, 6-0
Sofia Kenin (USA x4) bt Irina Maria Bara (ROU) 6-2, 6-0
Petra Kvitova (CZE x7) bt Leylah Fernandez (CAN) 7-5, 6-3
Zhang Shuai (CHN) bt Clara Burel (FRA) 7-6 (7/2), 7-5
Laura Siegemund (GER) bt Petra Martic (CRO x13) 6-7 (5/7), 6-3, 6-0
Paula Badosa (ESP) bt Jelena Ostapenko (LAT) 6-4, 6-3