In a much tighter match than the scoreline suggested, both women faced eight break points but Robson's battling qualities saw her through as she saved seven to Danilidou's four.
Robson, seeded eighth after cracking the top 60 in recent weeks, was less consistent than her opponent with the first serve but used her power from the baseline to unleash what are now trademark winners past the 30-year-old Greek, a former world number 14 who has since dropped outside the top 100.
Robson edged a tight first set that saw her defend a sole break, before racing 3-0 up in the second, although Danilidou staged a mini-fightback that saw her claw one game back against serve.
But the Londoner, who has developed a newfound resolve since the summer, showed great fight under pressure and broke again off a lengthy battle at deuce before closing it out with a nervy hold in one hour and 36 minutes.
The 18-year-old Briton, the youngest woman in the top 100, will next face Chinese qualifier Zhou Yimiao.
Fellow Brit Heather Watson made it through to the second round on Monday, as did number one seed Sam Stosur of Australia.
2-Zheng Jie (China) beat Maria Jose Martinez Sanchez (Spain) 4-6 7-5 6-4
Tamarine Tanasugarn (Thailand) beat 5-Yaroslava Shvedova (Kazakhstan) 6-3 6-2
6-Anabel Medina Garrigues (Spain) beat Camila Giorgi (Italy) 3-6 6-2 6-2
7-Chanelle Scheepers (South Africa) beat Alexa Glatch (U.S.) 6-4 6-1
Zhou Yimiao (China) beat Kurumi Nara (Japan) 7-5 6-4
Virginie Razzano (France) beat Nudnida Luangnam (Thailand) 6-3 6-3
Mirjana Lucic (Croatia) beat Luksika Kumkhum (Thailand) 4-6 6-3 6-4