The No. 2-ranked Nadal is aiming to go one better than Roger Federer and win a men's-record 21st Grand Slam singles title.
He didn't play for Spain at the ATP Cup last week and has been troubled by stiffness in his lower back.
"Well, it's been a tough 15 days for me. I had some issues with the back,'' Nadal said in his postmatch interview. "I needed to survive today. That's what I did.
"Straight sets -- that's what I need.''
Nadal said he needed to change the motion of his serve "a little bit" to help with the discomfort, adding that tomorrow is a day off before another match on Thursday.
"I need to go day-to-day and just try to stay positive," Nadal said. "Of course every day that I am trying to stay here longer is a day with a chance to get better finally. ... Trying to do all the things possible to be ready [to] compete, for what I came here."
Facing a break point late in the third set, Nadal sprinted to his left and hit a running, around-the-post forehand that landed just inside the line.
Djere, ranked 56th, shrugged, his head dropped, and he might have wondered what more he needed to do to win a point against the 20-time major champion. He had three break points when Nadal was serving for the second set, and didn't get close to cashing in one.
When Djere, going for everything, double-faulted to open what became the last game of the match, a woman in the crowd at Rod Laver Arena yelled out, ''It's all right, Laslo.''
Easy for her to say.
While Nadal has been slowly building into the tournament, Russia's ATP Cup-winning teammates have been on a roll.
Tsitsipas has a lot of support within Melbourne's large Greek community and felt at home in an extended on-court interview.
Tsitsipas says "I honestly wasn't expecting it to come so easily. It was a great match from my side,"
There will be fans with divided loyalties in his next match against Australian wild-card entry Thanasi Kokkinakis in the second round.
Kokkinakis has Greek heritage and also has a big following despite his ranking dropping to 267 following struggles with illness and injuries.
He hit 19 aces in a 6-4, 6-1, 6-1 win over Kwon Soon-woo.
ESPN's D'Arcy Maine and The Associated Press contributed to this report.