Nadal, whose 12 titles at Roland Garros are more than any player has ever won at a single Grand Slam, required just over two hours on Court Philippe-Chatrier to defeat his Belarusian opponent 6-4 6-4 6-2.
Thiem, who lifted his maiden major trophy earlier this month at the US Open, beat fellow Slam champion Cilic 6-4, 6-3, 6-3 in two hours and six minutes – spending just one minute longer on court than Spaniard Nadal.
That victory begins what could be a tough road to a third consecutive final in Paris for the Austrian.
In the women’s draw, Serena Williams also got up and running with a straight-sets win, defeating fellow American Kristie Ahn 7-6 (2) 6-0.
After a closely-contested opening set, Williams, 38, breezed through the second to give her considerable momentum in her bid for a 24th Grand Slam title – a feat that would match the record set by Margaret Court.
Meanwhile, the notable exits on Monday included fourth seed Daniil Medvedev – beaten by Marton Fucsovics in four sets – and eight seed Gael Monfils in the men’s draw, along with Fabio Fognini (seeded 14th) and teenage hotshot Felix Auger Aliassime.
In the women’s draw, 12th seed Madison Keyes was defeated, as was Marketa Vondrousova (seeded 15th).
It was another poor day for the Britons, too, with Cam Norrie beaten in five sets by lucky loser Daniel Galan, and Liam Broady ousted in four by Jiri Vesely.
Those exits followed defeats for wild card Andy Murray, Dan Evans and Johanna Konta on Sunday, with Heather Watson the sole remaining Briton in the clay-court major, which was delayed by five months due to the coronavirus pandemic and is taking place with just 1,000 spectators permitted in the grounds each day.