It proved to be a difficult opening set for Frenchmen Tsonga, who was broken once by Klizan on his way to securing a 6-4 success.
Klizan, currently ranked 52nd in the world, looked to be heading for a routine second-set victory when he secured three break points at the first time of asking.
However, Tsonga responded positively to win that game, before going on to win five of the next six available to level up the scores at one set apiece.
Tsonga was now expected to take control of the contest against an opponent that he had never met before competitively. The opposite happened though, as Slovakian Klizan claimed a double break of Tsonga's serve to win through 6-1 and regain his lead.
Another double break followed in the fourth set as he opened up a 4-1 lead. Tsonga managed to gain one of those back, but his effort was rendered meaningless as Klizan kept his nerve to hold his next serve, thus booking his place in round three.