The three-time grand slam champion is still recovering from the knee injury that played a big part in his second-round exit to Tennys Sandgren in Melbourne.
It was not the best start for Wawrinka, but he soon found his groove to secure a 4-6 6-2 6-3 victory over his Slovakian opponent.
Wawrinka fired down 16 aces and dropped just eight of his first-service points as Klizan failed to maintain his first-set standard.
Next up for Wawrinka is sixth seed Viktor Troicki, who was a 7-6 (7-4) 1-6 7-6 (7-5) victor over Denis Istomin, while Gilles Muller (3) was also taken the distance in his beating of Andreas Seppi.
However, fellow seeds Adrian Mannarino and Joao Sousa both bit the dust, the former was a 6-7 (4-7) 6-3 6-1 loser against Marcos Baghdatis, and the latter fell 7-5 7-6 (8-6) to Maximilian Marterer. Marius Copil was a comfortable 6-2 6-2 winner against Blaz Kavcic.
In Montpellier, Richard Gasquet's quest to add to his four career Open Sud de France titles continued in a 7-6 (7-3) 5-7 6-3 victory over French compatriot Pierre-Hugues Herbert.
Second seed Lucas Pouille had little trouble in seeing off Carlos Taberner 6-1 6-2, but Jo-Wilfried Tsonga was taken the distance by Nicolas Mahut in a match where each set went to a tie-break.
Benoit Paire added to the French contingent that reached the last eight with a 6-4 7-6 (7-4) triumph over John Millman.
Damir Dzumhur, the fourth seed, defeated Ruben Bemelmans, while Karen Khachanov overcame Ricardas Berankis in straight sets - the second of which came in a mammoth tie-break that the Russian won 13-11.