He beat off competition from Hereford gaffer Turner, Southend chief Tilson and Hutchings of Walsall.
Crewe recorded four wins out of five in the league in February with victories over Brighton, Yeovil, Huddersfield and Southend.
The Gresty Road side scored 10 goals in the process and now currently sit 20th in the table, one point from the drop zone.
"I have been very pleased. The players have responded well," Thordarson told Sky Sports News. "They worked hard and committed themselves and under difficult circumstances have taken themselves forward in a positive manner.
"We have picked up some good results which have taken us out of the relegation places at the moment.
"I always felt I could help them. I always thought I could get them out of relegation but we haven't completed the job, there is a long way to go yet."