There were four heats, consisting of a maximum five runs. The two best-placed riders after three of those runs advanced to the semi-finals, with the next two going through after completing all runs.
Australia's Willoughby found himself behind as the Netherlands' Twan van Gendt and defending Olympic champion Maris Strombergs advanced with two runs to spare.
A second place in the fourth run saw Willoughby certain of his place, and he stayed out of trouble on the fifth run to focus on the last 16.
'A couple of mistakes early cost me, but I was still pretty safe so I saved the energy for tomorrow,' Willoughby said.
It's not a big deal. It was a good little learning curve for me and I'll be better for it tomorrow.Sam Willoughby
'I only did one extra [run], so I'm fine. It's not a big deal. It was a good little learning curve for me and I'll be better for it tomorrow.'
Strombergs finished second, fifth and first in his three runs.
The Latvian said: 'I never thought it was going to be that hard to come in as Olympic champion. It's a lot of pressure and when you start thinking about it, it's really hard to focus on yourself and what you really want to go out there and do.'
Strombergs was pleased to advance ahead of Willoughby. He added: 'When I was in the gate I was pretty sure I was going to have to do five [runs].
'When I crossed the finish line [on the third run] I looked up and saw I was second, I was relieved.'
Raymond van der Biezen of Netherlands, the fastest qualifier, showed he would be a threat by advancing with wins in all three runs.
He said: 'I'm stoked. I know what I can do. I know what I'm capable of. I just need to be there at the right time. I'm happy with that.'
Latvia's Edzus Treimanis advanced from heat one behind the Dutchman. World time-trial champion Connor Fields of the USA advanced from heat two with a 100 per cent record, with Great Britain's Liam Phillips following him through.
Carlos Oquendo Zabala of Colombia, Quentin Caleyron of France, Australia's Khalen Young, Andres Jimenez Caicedo of Colombia, Rihards Veide of Latvia, USA's David Herman and Switzerland's Roger Rinderknecht also progressed.