The Premier League outfit were 2-0 inside the first 30 minutes but surged back into the contest and took the lead through goals from Alexis Mac Allister (two), Ronald Araujo and Florian Wirtz.
Villarreal found the back of the net late, but it proved a mere consolation as the Reds held on to take maximum points.
In quotes published on the Liverpool website, Nyoni said: "There was no panic at all. Everyone believes in each other and I feel like when the team morale is like this, none of us can panic and we believe in our quality.
"You saw in the second half how we changed the game and upped the tempo a bit more, which is what we needed to do. And scored the goals."
It was a first start in Europe's elite competition for the 19-year-old Nyoni, and the teenager was grateful to be given the responsibility while learning form his more experienced teammates.
He added: "It's special. I've had to wait but I feel like it's all part of the process. I've learned a lot over the last few years, even just being on the sideline watching the rest.
"My teammates help me, especially the senior players, they help me a lot when I'm on the pitch and they try to make my job as easy as possible. I can't ask for more."
Enzo Maresca has made it his mission to help Phil Foden rediscover his best form and re-establish himself as one of England's leading players after the Manchester City midfielder ended his goal drought in spectacular fashion.
Vincent Kompany has played down fears of serious injuries to Luis Diaz and Min-jae Kim after Bayern Munich's 2-2 draw with Arsenal in the Champions League on Wednesday.
Jose Mourinho has praised Real Madrid's squad depth after their 4-1 Champions League win over RB Leipzig but insisted his team must improve before Sunday's El Clásico against FC Barcelona.
Virgil van Dijk believes Rio Ngumoha has a bright future at Liverpool after the teenage winger played a key role in their thrilling 4-3 Champions League victory over Villarreal at Anfield.
