Mayer, Starace shocked in Bucharest
The German world number 21 (pictured) earned plenty of chances against the 31-year-old, but converted just three of his 10 break points while Malisse was ruthless in taking four of his five opportunities.
"I played well," said Malisse. "I think we played a good match and I’m very happy to win at the end. It feels good.
"This year’s been tough for me and it’s the first time I’ll play quarter-finals this year. Physically I feel good after a long match. I worked hard last week and I’m happy it’s paying off."
Italy's Potito Starace also fell to a surprise defeat, losing 6-3 6-4 to Hungarian qualifier Attila Balazs, a player ranked just 450th in the world. Balazs had already sprung a surprise by knocking Lukas Lacko out in the opening round.
Germany's Daniel Brands also made the third round thanks to a 7-6(3) 6-3 win over Estonia's Jurgen Zopp.