Preston fought back with a battling second-half display to book their place in the fourth round of the FA Cup.
It was a dull first half on a cold day watched by the second lowest crowd of the season of 8,380.
Crewe's Chris McCready had a diving header after eight minutes which went wide of the right-hand post after a cross and Michael Higdon was put clean through on goal after 12 minutes, but Preston keeper Carlo Nash ran out of his box swiftly and tackled him and cleared the ball.
Chris Sedgwick had a left-foot shot saved by 20-year-old Crewe keeper Stuart Tomlinson who was making his first start and Lewis Neal had a run and shot as he fell over which was saved by Tomlinson.
Billy Jones made it 1-0 to Crewe after 34 minutes when a free-kick by Kenny Lunt, the Crewe captain, was mis-punched by Nash as the ball went over his head. Lucketti headed clear on the line but only to Jones who headed home from two yards.
Just before half-time David Nugent shot wide but Crewe deserved their half-time lead.
Callum Davidson had a 25-yard left-foot shot deflected for a corner and Billy Jones had a header saved by Nash from a Lunt free-kick.
After 61 minutes Chris Sedgwick was tripped just outside the box and Graham Alexander hit a 25-yard free-kick which was deflected off the top of the wall to the keeper's left and into the back of the net to make it 1-1.
Sedgwick had another rasping shot which went wide to the left of the post.
Davis and Neal had shots well saved by Tomlinson and sub Dave Hibbert went close with ten minutes to go, but it was Sedgwick with five minutes left who got Preston's winner with a 20-yard volley to the keeper's left after sub Danny Dichio had headed the ball on.