The top seed and defending champion took more than two hours to overcome the world number 199 after having received a bye through to the second round.
After fighting back from a sluggish second set in which his percentage of points won on first serve dropped from 75 per cent to 57 per cent, Murray regained his poise in the third set to beat the 23-year-old Australian and secure his safe progression.
"It was a real high quality match," Murray said in his courtside interview. "I expected a tough match. He is a lot better than his ranking, that is for sure."
Murray revealed that former world number one Lendl, who was due to arrive in Brisbane on Friday, had not even considered the possibility of the Scot losing to Millman.
"He was obviously expecting me to come through, but it was very nearly a different story. The depth of the game now is so, so strong. None of the matches are easy. It was a bit different back in Ivan's day where they could kind of cruise a little bit the first couple rounds.
"I had seen John's name before, but I hadn't seen him play. I think it's important when you don't know how someone plays to find out about their character a little bit.
"He's very consistent...he can play quite a long way behind the baseline and doesn't take too many risks. And then, you know, he was going for absolute bombs and making them," he added.
The world number three will next take on Uzbekistan’s Denis Istomin, who defeated former number one, 31-year-old Lleyton Hewitt, 7-5 7-5.
Murray, who won the tournament last year before reaching the semi-finals of the Australian Open, is continuing his warm-up for the first Grand Slam of the year in Melbourne, which gets under way on January 14.
Meanwhile, Bulgaria’s Grigor Dimitrov beat Milos Raonic 6-3 6-4 to send the second seed packing at the second-round stage.
The world number 13 failed to win a single point on the first serve of his opponent as he was unable to put any pressure on the world number 48.
Dimitrov won just three points on the Canadian’s first serve, but won all 26 of his first-serve points en route to the straight-sets win. The Bulgarian will next take on seventh seed Jurgen Melzer of Austria.
"I just didn't take my chances. Should have been up an early break in the first and the second sets," said Hewitt.
"I just tried scrapping around to hang in as long as I could, but it was just one of those days. Two matches has been an absolute bonus this week, but it would have been nice to have a few more."
ATP Brisbane results
1-Andy Murray (Britain) beat John Millman (Australia) 6-1 5-7 6-3
Grigor Dimitrov (Bulgaria) beat 2-Milos Raonic (Canada) 6-3 6-4
7-Jurgen Melzer (Austria) beat David Goffin (Belgium) 6-4 7-6(4)
Denis Istomin (Uzbekistan) beat Lleyton Hewitt (Australia) 7-5 7-5
ATP Doha results
Nikolay Davydenko (Russia) beat Simone Bolelli (Italy) 6-1 6-1
1-David Ferrer (Spain) beat Paolo Lorenzi (Italy) 6-3 6-0
2-Richard Gasquet (France) beat Lukas Lacko (Slovakia) 6-1 6-3
Daniel Brands (Germany) beat Gael Monfils (France) 6-1 7-5
ATP Chennai results
1-Tomas Berdych (Czech Republic) beat Somdev Devvarman (India) 6-3 6-1
3-Marin Cilic (Croatia) beat Sergiy Stakhovsky (Ukraine) 4-6 6-2 6-3
5-Benoit Paire (France) beat Dudi Sela (Israel) 6-1 6-0
Roberto Bautista (Spain) beat Matthias Bachinger (Germany) 7-6(4) 6-3