American Eaton dominated the sport in recent years, winning the last two Olympic and world titles.
Mayer, silver medallist at the Olympics last year, produced personal bests in the 100 and 400 metres and still has two of his strongest events to come on Saturday - the pole vault and 110 metres hurdles.
The European record holder produced finished fourth in the 100 metres in 10.70 seconds, the first of the 10 disciplines.
The 25-year-old also performed well in the shot put, where he threw a season's best 15.72 metres to finish second, and was fourth in the long jump with 7.52 metres.
After coming joint fifth in the high jump, he finished his day by running 48.26 seconds in the 400 metres, the seventh best overall time, to lead with 4,478 points from Germans Kai Kazmirek, on 4,421, and Rico Freimuth on 4,361.
"I am a perfectionist. There are still many things I have to look at," Mayer said. "There is still a long way to go until the last event. Everybody keeps telling me that I am the favourite, but before coming here I was not the favourite."
Kazmirek, eighth after the shot put, leapt up to second after winning both the high jump (2.11 metres) and the 400 metres (47.19 seconds).
Canadian Damian Warner, bronze medallist in Rio, began strongly by winning the 100 metres in 10.50 metres but slipped down the field after struggling in the shot, where he threw a modest 13.45 metres.
He crept up to fourth again after producing the second best overall time in the 400 metres of 47.47 seconds.
(Editing by Ed Osmond)
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