The incident involving Suarez came after a frustrating night for both the player and his team who lost 1-0 with Socceroos goalkeeper Mark Schwarzer keeping a clean sheet for the hosts at Craven Cottage.
Fulham won thanks to a late Clint Dempsey goal, after Liverpool had been reduced to 10 men when Jay Spearing was controversially sent off 18 minutes from time.
Dalglish said Suarez enjoyed his and the club’s support and that others should follow their example rather than criticise him constantly.
Suarez, who was charged by the Football Association (FA) in November with racially abusing Patrice Evra, had a goal disallowed and also felt he should have had a penalty.
“We’ll look after Luis as best we can, and I think it’s about time he got a bit of protection from some people,” said Dalglish, who admitted he hadn’t seen the incident involving Suarez and the Fulham fans.
“His integrity is impeccable. He can look in the mirror and go to bed at night and sleep soundly. If other people can do that then fine. All we ask is to be fair, is to be honest.”
Dalglish, whose side had won their previous six away matches, said they would consider appealing against referee Kevin Friend’s straight red card to Spearing.
The Liverpool midfielder received the card for a tackle on Mousa Dembele where he won the ball but caught his opponent when he followed through.
“Jay never had another thought in his mind but to win the ball, and he did win it,” Dalglish said. “As long as they are consistent, it’s not a problem. But I don’t think we were going to get anything tonight, were we?”
Fulham manager Martin Jol felt the red card was justified.
“I saw it from three angles and it was a bad one. He could have broken his ankle. I think it was a sending off,” the Dutchman said.
There was also a case for Fulham’s Swiss defender Philippe Senderos to receive a second yellow card, with his first coming after bringing down Charlie Adam on the edge of the penalty area, which Liverpool also felt was a penalty.
Dempsey, whose goal was his fourth of the season, admitted the sending off changed the game.
“We had more space and time on the ball after the sending off,” said the American.
His goal came five minutes from full-time after Jose Reina spilled a shot from former Liverpool player Danny Murphy.
“If you win 1-0 and keep a clean sheet it’s always pleasing. The sending off us helped us with giving us more space and Clint then scored the goal,” Jol said.