Chilean Tabilo, ranked 89th in the world, had beaten the 24-time grand slam singles champion in both of their previous meetings, which came on clay in Rome last year and Monte Carlo in April.
But top seed Djokovic served 13 aces and faced just one break point on his way to a 7-6 (7-3) 6-1 victory on the hard courts of the Greek capital, where elite-level tennis returned for the first time since 1994, to reach the 225th tour-level quarter-final of his career.
The Telekom Center crowd offered constant support for the Serb, who moved with his family to Athens earlier this year.
"It feels really like home playing in Athens," Djokovic said in an on-court interview.
"More than recognition for my tennis achievements, I felt that people here approached me in a friendly and humane way, and that touched my heart."
A tight tussle in the opening set saw only two break points - both in Tabilo's opening service game - before Djokovic dominated the tiebreak.
The 38-year-old clinched the first break of the contest in the fourth game of the second set, which proved to be the decisive moment as Djokovic clinched victory in 99 minutes.
"Playing against Tabilo, who I had never won against - we played two times on clay, this year and last year and he won both of these matches - I was more under tension before the match than some other matches and I really tried to draw the energy from the crowd," Djokovic added.
In other results on Tuesday, there were wins for American pair Sebastian Korda and Marcos Giron, German Yannick Hanfmann and Nuno Borges of Portugal.
In Metz, meantime, Australia's Aleksandar Vukic opened his campaign with a 3-6 7-6 (8-6) 7-6 (10-8) marathon win over eighth-seeded Frenchman Corentin Moutet.
Next up for the 29-year-old Sydneysider after his two-hour 45-minute battle is Italian Matteo Berrettini.
Elsewhere, Italy's Lorenzo Sonego, Belgian Alexander Blockx and Brit Cameron Norrie were also victorious in three sets, while Daniel Altmaier of Germany recorded an easier win.
With AAP.
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