The defender scored the goal in the second half that pulled Everton level at 1-1 following Nabil Fekir's sixth-minute penalty but he couldn't prevent Bertrand Traore from scoring what eventually proved to be the winner for the French club.
The result means that Ronald Koeman's men have made the worst start to a Europa League group stage campaign than any English club before them and it has drastically reduced their chances of progressing to the knockout stages.
“It's going to be difficult [to progress]," Williams admitted in the tunnel after the game. "We wanted to win tonight. We didn't get any points but I thought the lads responded well when we went one down.
“But we've come away with nothing. I thought we fought well as a team."
The Welshman was handed the captain's armband in the absence of Phl Jagielka and Leighton Baines but was fortunate to escape a red card for a second-half melée that involved supporters in the crowd at one point as the contest boiled over.
Williams was captured by television cameras exchanging blows with a Lyon player but only received a booking and stayed on the pitch to equalise shortly afterwards.
Asked to describe the incident, which may well land him in retrospective hot water with Uefa, Williams dismissed it as a consequence of raw emption and the desire to get back into the game.
“That's what happens. It's football. I think it is what it is… it is what it is.
“It's high emotions. We want to win the game, they want to win the game and stuff happens and this was one of those times, I suppose.
“The pressure that you put on yourself is bigger than the pressure from outside,“ the defender continued when asked if he and the players were feeling the strain from Everton's poor run of results.
“We want to show fight for ourselves more than anything, for the club, and for the fans, especially at home. We're disappointed not to get a result tonight because I thought we had a good attitude, we went about it [right].
“We had a lot of chances — more than them — and it's really disappointing when we got 1-1 to concede another goal so quickly.
“[The Arsenal game] is a big one. They're all big, especially when you play against a top team at home. We need a win and we need a good performance.”