It was the away side who started the better and they might have been awarded a penalty after three minutes when Yakubu was seemingly felled in the box by Angel Rangel, but no decision was forthcoming.
Scott Dann might have sent Blackburn into the lead minutes later, but he headed over from inside the box.
Swansea's first real chance of the opening period fell to Sigurdsson after 15 minutes, but his effort from outside the box drifted over the bar.
Swansea might have been awarded a penalty after 25 minutes when Morten Gamst-Pedersen appeared to fell Sigurdsson in the box, but the referee waved play on.
Grant Hanley might have headed Blackburn into the lead after 26 minutes, before Yakubu came close for the away side on the 30-minute mark.
Both sides had chances as the half developed, and Swans midfielder Joe Allen cleared the bar with an effort after 33 minutes.
Swansea took the lead after 37 minutes, however, when Sigurdsson curled a fine effort past Rovers goalkeeper Paul Robinson.
Danny Graham might have added the home side's second five minutes before the break, but Robinson made a smart save.
The second did come, however, moments before the break when Dyer converted from close range.