Dyer was carried off on a stretcher from Carrow Road just before half-time, with a "a suspected fractured ankle", after he fell awkwardly in a challenge with Norwich defender Sebastien Bassong, having earlier put the visitors in front on 12 minutes.
Gary Hooper cracked in 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 Michel Vorm would have won the match for the Canaries.
Laudrup felt the loss of the energetic midfielder had a major impact on his team, which had been through a gruelling schedule following their midweek Europa League tie in Switzerland.
"I think the game changed in two ways with the injury. Things like this are part of football, even though we don't know how bad it is yet, but everybody could see it was not good when you get carried out like that," said Laudrup, whose squad lost defender Neil Taylor for most of last season when he fractured his ankle in three places.
"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 set-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.
"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 today.
"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."
Swansea confirmed on their Twitter feed Dyer had been taken to hospital with "a suspected fractured ankle".
Laudrup hopes the long-term prognosis will offer some positive news.
"You just go from playing one second to being out for you don't know how long," the Swansea boss added.
"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."