Austrian third seed Dominic Thiem needed less than an hour to brush past American qualifier Marcos Giron 6-1, 6-3, while Matteo Berrettini won against his fellow Italian Fabio Fognini 6-3, 6-4.
Meanwhile, world number seven Andrey Rublev had to work hard on his Madrid Open debut on Tuesday before booking a place in the third round with a 6-7 (5), 6-3, 6-4 win over Tommy Paul, Xinhua reports.
Rublev had problems adapting to the speed on the ball on the Madrid clay, where conditions are made a bit harder by the altitude of the Spanish capital, but is on track to continue his record of reaching at least the quarterfinals of every tournament he has played in all season.
In the first round, No. 9 Roberto Bautista Agut needed three sets before edging past Marco Cecchinato 6-2, 6-7(3), 7-5, and 11th seed Denis Shapovalov beat Dusan Lajovic 6-1, 6-3.
Qualifier Federico Dabonic produced a surprise by knocking out Spanish No. 10 seed, Pablo Carreno Busta, coming back from a set down to win 3-6, 6-4, 6-3.
In the women's tournament, No. 3 seed and two-time winner Simone Halep lost 4-6, 7-5, 7-5 to Elise Mertens to crash out of Madrid before the quarterfinals for the first time since 2015.
The Romanian struggled to find her game throughout the match, and ultimately 48 unforced errors were the difference in a close match.