Dyer was carried off on a stretcher from Carrow Road just before half-time, with 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.
However, a statement on Swansea’s Twitter page last night struck a more upbeat, if cautious, note.
“Initial x-ray shows no obvious break to Dyer’s ankle. However, he will have further scans tomorrow before final diagnosis.”
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 Michael Vorm would have won the match for the Canaries.
Laudrup felt the loss of Dyer had a major impact on 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,” said Laudrup.
“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.
“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.”
Norwich, meanwhile, will probably have felt they did enough to take all three points to build on their win at West Brom last weekend.
A header from defender Michael Turner hit the crossbar and was somehow deflected over off Jordi Amat before Vorm pulled off a fine reaction save to turn around Hooper’s goalbound effort.
Canaries boss Chris Hughton believes there is more to come from the former Celtic forward, signed in a £5 million summer transfer.
He said: “It was a fantastic and very special goal.
“Gary is a finisher, technically a good striker of the ball so always gives himself a chance. We needed someone to produce something and thankfully Gary did.”
NORWICH: Ruddy, Whittaker, Turner, Bassong, Olsson, Redmond (Murphy 83), Howson, Johnson, Fer, Hooper, Elmander (Becchio 76).
SWANSEA: Vorm, Rangel, Amat, Williams, Ben Davies, Dyer (Pozuelo 44), Shelvey, de Guzman, Canas, Hernandez (Lamah 83), Michu (Bony 75).
Referee: Michael Oliver.