Dyer was carried off on a stretcher from Carrow Road just before half-time of the Barclays Premier League match when he fell awkwardly in a challenge with Norwich defender Sebastien Bassong, having earlier put the visitors in front on 12 minutes.
The 26-year-old was taken to hospital, where an initial X-ray showed "no obvious break", but the club medical staff are set to assess the full prognosis over the coming days.
"You just go from playing one second to being out for you don't know how long," said Laudrup, whose squad had to do without defender Neil Taylor for most of last season when he fractured his ankle in three places.
"We will wait to see the X-ray and then in a couple of days have an MRI scan to know if it is bad or really bad.
"Let's wait and see, hoping that if it is bad, it is not too bad."
Laudrup was in no doubt the injury to Dyer unsettled his side in a match where they had looked in complete control.
Norwich forward Gary Hooper went on to crack home a superb 25-yard effort to equalise during first-half stoppage-time, chesting the ball down and firing it back over the goalkeeper - and - but for a fine late save from Michael Vorm - the £5million signing from Celtic would have won the match for the Canaries.
Laudrup felt the loss of the energetic midfielder had a major impact on the dynamic of his team, which had been through a gruelling schedule following their midweek Europa League tie in Switzerland.
"Everybody could see it was not good when you get carried out like that," he said
"We are all human, the team had 10 minutes where they suffered a bit and tactically we also lost our main threat with the runs from the second line, like when he had scored the goal.
"Even though it was a great goal to equalise just before half-time, then it is difficult to come back to the dressing room, even in the first 10 minutes of the second half, we suffered a bit, especially at pieces.
"After that, the game was a lot of play in midfield, with both teams trying to get through, but not really creating that many chances."
Laudrup added: "In the end, 1-1 I think at the end of a very difficult run of games, we can look at the positive things, even if it is a bit difficult with the injury.
"We have reached 20 points now and are among the last 32 in Europe for the first time ever and will see on Monday what the draw will bring us."
Norwich, meanwhile, will probably have felt they did enough to go on and take all three points.
A header from defender Michael Turner hit the crossbar and then was somehow deflected over off Jordi Amat before Vorm pulled off a fine reaction save to turn away Hooper's goalbound effort.
Norwich boss Chris Hughton was pleased to have seen his side edge a small step further clear from trouble following a difficult opening to the campaign.
Hughton said: "We are just pleased that at this moment we are on a decent run and have been able to pick up a few points, but we have to keep that going."
Norwich have announced forward Carlton Morris, who turned 18 on Monday, has been given a two-and-half year professional contract which comes on the back of some impressive displays in the FA Youth Cup winning side last season.