The Swans displayed commendable battling qualities at Anfield on Sunday, twice coming from behind to haul themselves level.
They were, however, to be edged out in a seven-goal thriller to remain in the bottom half of the Premier League table - four points above the drop zone.
Williams, though, believes Garry Monk's side proved what they are capable of on Merseyside and will enjoy success in the future if they can maintain those performance levels.
The Swans skipper said in the South Wales Evening Post: "We conceded four goals, which is never nice, but there are a lot of positives to take from it.
"We showed character to come back so many times away from home, which I'm not sure too many teams would do.
"Maybe before we wouldn't have had the belief that we could have come back. Now we do and there were some good performances out there."
Williams is now looking for Swansea to carry their renewed attacking threat into a testing trip to Italy on Thursday, with a last-32 encounter against Napoli intriguingly poised after a goalless first leg at the Liberty Stadium.
The Wales international added: "We know it's going to be a different game in Napoli to the one we played the other day and it will be tough for obvious reasons.
"We feel like we need to go there and enjoy it. To play in an atmosphere like that is something that maybe we're not used to.
"But we want to progress in this competition and feel that we've got enough quality in the team to do that.
"We need to play our game and stamp our authority on games, although we know we've got to be a little bit cautious away.
"We didn't score against Napoli at home, but it wasn't the fact we didn't create anything at home, we just couldn't get the ball in the net for whatever reason.
"We know we can score goals, we've been attacking very well now in the last few weeks. We're getting more players forward now with width and we're confident."