The victor of that match could face off against top seed Rafael Nadal, who plays Spaniard Roberto Carballes Baena in his first match of the tournament later on Wednesday.
Djokovic, a 12-time Grand Slam champion, has lost his touch of the old ever since an elbow injury compelled his withdrawal from Wimbledon past year.
Djokovic chose to play in Barcelona after losing to Dominic Thiem in the third round of the Monte Carlo Masters. The five-time victor on the ATP Tour won that opening game on Djokovic's serve with a fierce forehand down the line and, after surviving a break point to move 3-1 in front, Klizan stretched his lead yet further.
The Mallorcan responded by breaking twice to take a 4-2 lead and went on to claim the first set.
A second break for the Slovak followed soon after as he moved 4-1 up and a series of outrageous drop-shots helped him move a set ahead after just 32 minutes.
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The top-ranked Spaniard has won 38 consecutive sets on clay, extending his own record.
Nadal will next face Guillermo Garcia Lopez of Spain, who defeated Kei Nishikori after the Japanese player retired because of an injury.
The 28-year-old, who lost to Rafa Nadal in the Monte Carlo Masters final on Sunday, withdrew after losing the first set 6-3.
Nadal, coming off his 11th Monte Carlo title, trailed early in his match but closed it out to reach the third round.
Fatigue, and pain in his troublesome right wrist, appeared to prompt the withdrawal from Japans world number 22, who told reporters after the match: "I started feeling discomfort again during the first set and I did not want to take any risks". He missed last year's U.S. Open and the Australian Open at the start of 2018 after injuring the same wrist in August.