The three-time Grand Slam champion fought back from a break down in the final set to overhaul the 35th-ranked Slovak 4-6, 6-4, 7-5, 4-6, 6-4 in 3hr 24min on Margaret Court Arena.
There was a dramatic moment in the final set when Wawrinka fired a volley at a defenceless Klizan, who had conceded the point after a rally and was standing at the net.
The Swiss fourth seed quickly jumped over the net to offer his apologies as Klizan sunk to his haunches in pain.
"That was, for sure, a tough shot for him. The thing is there are some players they fake you, and you stop playing and then they just put up the racquet," Wawrinka said.
"So for me I already miss some easy ones. For me the most important thing was to run straight over to him.
"I just wanted to make sure he was okay. I realized he was okay.
"I'm sorry I touch him in the wrong place. But I think it was okay at the end."
Klizan preferred not to pass judgement on Wawrinka but he said the Swiss could have hit his volley anywhere on the court.
"I stopped playing. He could play anywhere, the whole court," the Slovak said. "He could play anywhere on the court and he hit me."
Despite the incident, Wawrinka was relieved to get through a difficult first-up match to stay alive at the year's first major.
"It's most important to get through that match, to win. For sure it's a big relief, especially when you are a break down in the fifth, 4-3, 40-15, I'm really happy to get through," he said.
Wawrinka, who beat Rafael Nadal to win the 2014 Australian Open, will next play American Steve Johnson.
The Swiss hit 21 aces and 13 winners with his signature single-handed backhand, but made 43 unforced errors as he struggled to contain left-hander Klizan's powerful forehand.
Wawrinka maintained his record of never having lost in the first round at his 12th Australian Open campaign.
Also on the first day of the Australian Open, Swiss 17-time Grand Slam champion Roger Federer won his first match after coming off a injury-blighted 2016.
Federer downed his one-time junior rival Jurgen Melzer of Austria 7-5, 3-6, 6-2, 6-2 in 2hr 6min on Rod Laver Arena.
The Swiss 17th seed will now play American qualifier Noah Rubin in the second round.