Evans, the world number 43, had on Monday produced a first ATP Tour win on clay when he came through against Thiago Monteiro after a final-set tie-break.
The 26-year-old backed that up with a notable second-round triumph over Zverev, who posted an unexpected win against world number one Andy Murray at the Australian Open.
Evans took his first set-point opportunity against Zverev's serve and then rallied to level the second set at 3-3 having been broken by the 14th seed.
After seeing more break chances missed in game eight, Evans managed to fend off a recovery by the world number 35 as he held to then break Zverev again to love and seal victory in one hour and 23 minutes.
British number two Edmund had confidently defeated Frenchman Jeremy Chardy in round one, but could not get the better of Thiem on Tuesday as the Austrian world number nine won 6-1 6-4.
Thiem, who is seeded fourth, twice broke Edmund in the first set and then secured another crucial break to lead 3-2 in the second.
Edmund, the world number 42, could not then force two break chances of his own in the next game as Thiem went on to close out victory in one hour and 21 minutes to clinch his place in the third round.
Murray, meanwhile, is set to continue his comeback from an elbow injury on Wednesday when he takes on Bernard Tomic following the Australian's first-round win over German Dustin Brown.
Frenchman Richard Gasquet, who is returning to action following appendix surgery, 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, British number four Aljaz Bedene beat lucky loser Marius Copil 7-5 6-2 to move into the second round.
Copil took his place in the main draw following injury to Argentina's Federico Delbonis.
Bedene will next play Dutchman Robin Haase, the world number 46.
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, while Russian veteran Mikhail Youzhny went down 6-2 6-2 against Hungarian Marton Fucsovics and Germany's Florian Mayer retired when losing 6-4 3-0 against Ukranian Sergiy Stakhovsky.
