Davies was given a deadline of the end of September to claim the managers job and the home game against Plymouth on Tuesday could be the decider, but he deserves the job for his side's effort.
Eddie Lewis saw a right-wing left-foot corner which the keeper saved after seven minutes, but he was fouled by Brian O'Neil and Dean Ashton had a volley saved at the other end after 11 minutes.
Richard Cresswell had a shot from the right saved by the keeper after a deflection and then he had another shot parried by Ben Williams, who managed to pick the rebound up.
Lewis had a 30-yard central left-foot free-kick which hit the left-hand stanchion after Steve Foster had fouled Cresswell.
A Callum Davidson cross was handled by Crewe captain and star man Kenny Lunt.
A Graham Alexander right-foot short free-kick found Davidson, who was making his full league debut, and he hit a left-foot 20-yard shot into the centre of the net past two defenders and the keeper on the line, to make it 1-0 to Preston.
Right on half-time a Lunt free-kick from the right was met by a Foster eight-yard header and Andy Lonergan dived to his left to make a great save.
David Vaughan hit a free-kick which Lonergan pushed wide for right-wing corner and Andy Smith turned in the box to shoot straight at the keeper, left-footed from 15 yards on the hour mark.
Nineteen minutes from time Lynch was sent off for kicking Cochrane, being shown a straight red after just seven minutes on the pitch.
Steve Jones had a shot saved by Lonergan after he cut inside Youl Mawene late on, but the Lilywhites held on to take all three points.