The Australian came through 7-5 4-6 6-2 against the German world number 75 and will meet Murray on Wednesday as the Scot continues his recovery from an elbow injury.
British number two Kyle Edmund, though, is out after losing 6-1 6-4 against Dominic Thiem of Austria.
The number four seed will next meet another British player after Dan Evans produced a notable 6-4 6-4 win over Germany's Mischa Zverev to move into the last 16.
Frenchman Richard Gasquet, who is returning to action following appendix surgery, also failed to progress when the ninth seed lost in the second round against Yuichi Sugita, the Japanese lucky loser coming through 4-6 6-3 7-6 (7/3)
German eighth seed Alexander Zverev is also into the third round after his 7-6 (7/4) 4-6 6-4 victory over Nicolas Almagro of Spain, while Pablo Carreno-Busta, the seventh seed from Spain, defeated Italian Andreas Seppi 6-4 6-2.
Spain's David Ferrer, the 13th seed, is out after he lost 6-3 6-4 to Kevin Anderson of South Africa.
Brazilian Rogerio Dutra Silva saw off Argentina's Renzo Olivo 6-1 6-1 to set up a second-round match against nine-time Barcelona Open winner Rafael Nadal.
At the Hungarian Open in Budapest, Jiri Vesely beat fellow Czech player Borna Coric 7-6 (7/4) 6-4 to secure a date with top seed Lucas Pouille of France in round two.
Fifth seed Gilles Simon of France lost 6-4 6-1 to American Bjorn Fratangelo.
British number four Aljaz Bedene beat lucky loser Marius Copil 7-5 6-2 and will next play Robin Haase after the Dutch world number 46 beat eighth-seeded Argentinian Diego Sebastian Schwartzman 6-1 6-3.
Russian veteran Mikhail Youzhny went down 6-2 6-2 against Hungarian Marton Fucsovics, while Germany's Florian Mayer retired when losing 6-4 3-0 against Ukranian Sergiy Stakhovsky.
Read the original article on Eurosport: Tomic sees off Brown to book Barcelona date with Murray