The towering 37-year-old won the opening 12 games with minimum fuss to spark chatter of a rare Grand Slam main draw "triple bagel", which has been achieved by only five players in the professional era, the last being Sergi Bruguera in the 1993 French Open.
Cilic, the 2018 runner-up at Melbourne Park, saved a breakpoint at the start of the third set to remain on track but Altmaier suddenly clicked into gear.
The German made it 1-1, and in doing so snapped a 25-game losing streak stretching back to his 6-3, 6-0 defeat by Jaume Munar in the Adelaide International last week.
It was a much more evenly-contested match from there and Croatian Cilic had to rally from behind in the tiebreak to close out the win.
Fellow veteran and 2014 Australian Open champion Stan Wawrinka then joined Cilic in the second round, the Swiss battling from a set down to see off unseeded Serb Laslo Djere 5-7, 6-3, 6-4, 7-6(4).
The 40-year-old, who won his three Grand Slam titles during the golden era of Roger Federer, Rafa Nadal and Novak Djokovic, is making his final Melbourne Park appearance as a wild card after announcing that 2026 would be his last season on the tour.