The £22.7m signing from Ajax scored after he rounded keeper Asmir Begovic and unleashed a shot which was diverted by defender Andy Wilkinson against the post and into the goal.
Suarez's goal, which came 16 minutes after his introduction as a substitute - in addition to Raul Meireles's opening strike - helped lift the gloom at Liverpool following the departure of forward Fernando Torres to Chelsea for £50m.
Torres handed in a written transfer request last week to rock the Merseysiders, who brought in Suarez and £35m striker Andy Carroll as his replacements ahead of the January transfer window's closure on Monday.
"It has been a busy week and we've been in the newspapers more than we would have liked to have been," added Dalglish.
"It is great credit to the players that they have not let it get to them and they've gone out and been magnificent.
"The supporters realise that there is a great group here who want to compete for everything and get as much as we can from every game."
Liverpool were the better side - Pulis
While Suarez made an instant impact on his debut, Carroll remains a few weeks away from making his as he recovers from a thigh injury.
But Dalglish believes the fans' patience will be rewarded when Reds boss is able to pick Steven Gerrarrd, Suarez, Meireles and Carroll in the same team.
"You are starting to frighten some of the opposition," said Dalglish when asked about the prospect of being able to play the quartet together.
"They are four fantastic players. Andy is going to be a few weeks away from being fit but we bought him for five and a half years so a few weeks out of that won't do us any harm.
"Those four are really good footballers and they will complement each other very well."
The clean sheet against Stoke was Liverpool's third consecutive shutout and came after Dalglish started with a three-man central defence of Sotirios Kyrgiakos, Martin Skrtel and Daniel Agger to try to nullify the visitors' threat.
"They [Stoke] are a difficult team to play against - you have to compete against them first and then you have to get your own game going," said the Scot.
"Soto [Kyrgiakos] was brilliant for us. His presence and the way he commanded nearly everything which came in the box was great and when we were under pressure he was there as well.
"Three clean sheets is always a bonus and a help."
Stoke manager Tony Pulis conceded his team were beaten by a superior side.
"Liverpool were the better team and created the better chances," he said.
"We did okay for the first half but to concede just after half-time knocked us. But up until the second goal I thought we were in the game."
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