Chairman: Blackburn not in crisis
Williams also asked fans and shareholders for solidarity after Monday's decision which was described by Manchester United manager Sir Alex Ferguson as "absolutely ridiculous" and had fans and pundits scratching their heads in bewilderment.
"We are not in crisis - in fact, we are well placed in the table," Williams said in a statement.
"But we know only too well how tight the Premier League is this year and we are about to embark on a spell of six major first-team fixtures in three weeks."
Rovers are 13th, only five points off the top six after winning four of their last seven Premier League games, although they are also only five points above the relegation zone.
World Cup winner Diego Maradona's name was linked with the job but the club's Indian owners firmly ruled him out and first-team coach Steve Kean will be in charge for Saturday's match against rock-bottom West Ham United.
The Lancashire club were taken over by Indian poultry giant Venky's last month and the new owners said Allardyce's vision for the club did not fit in with theirs.
"We need solidarity in every area," added Williams, who described Kean as "out-and-out football man and a true professional."
"Football is a precarious business, fuelled by opinion, but I repeat that this is a time for everyone with a love for Blackburn Rovers to really show their support.
"We do not want negativity around the place. That could work to our disadvantage by playing into the hands of our upcoming opponents."
"This is not about individuals either. It is about the club - first, second and third," Williams added.
"The decision has been taken and we must move on."