Pat Fenlon quit as manager after Hibs' second defeat against their administration-hit rivals and another loss to Gary Locke's youthful side would be hard to stomach for the Easter Road faithful on Thursday night.
But Williams is focusing on repeating the hard work that has brought them rewards since Terry Butcher took over as manager.
Butcher and assistant Maurice Malpas have suffered just one defeat in their seven games in charge, which came against Celtic, and Williams has only conceded two goals during that spell. Sunday's 3-0 win over Kilmarnock was their biggest victory of the season.
Williams said: "It's important for us to keep the momentum going. It's more important for us to approach the game like any other game.
"We will approach it exactly the same way we approached Kilmarnock on Sunday, and make sure the performance and work ethic is there, and the game will take care of itself.
"If we were to get tied up in the revenge theory, it's never good to go into a game with a negative thought process like that.
"You play opponents that many times, you are bound to lose along the way. You would be forever running around trying to get revenge on people. That's no good.
"We have got some real good positives to focus on and that's what we will be doing."
Hibs fans, still wounded by their 5-1 Scottish Cup final defeat by Hearts in 2012, would find it hard to take another loss against a squad that is stretched to the limit by financial constraints and football sanctions.
But Williams said: "The joy of winning is much stronger than the fear of losing. That's something we are now developing.
"We are developing a momentum and a self-belief and confidence within that, and I think it's important that we carry that on.
"The boys have shown a real togetherness and team spirit in the last seven games and it's important we don't forget what we have done right.
"Obviously we understand the importance to the fans of a derby. We don't miss that at all, of course we understand that.
"We are looking to send people away happy and with a real belief that the tides have changed at Hibs."
The goalkeeper added: "We probably were in a position at the start of the season where our games were decided by one goal. So if we went behind, the belief wasn't really there that we could come back and win, although we did at Kilmarnock.
"Whereas now the belief is there that 90 minutes is a long time in football and if we have any setbacks we have the personnel to deal with that and turn the game around."