Grant McCann's side produced a perfect away performance at Gresty Road with goals in either half from Luke Molyneux and Harrison Biggins.
Rovers rode out some early pressure from the hosts, with Thimothee Lo-Tutala required to make a couple of saves as Crewe went in search of an early opener.
But once they had seen off those half-chances they took the lead against the run of play on 34 minutes. Hakeeb Adelakun and Matty Craig worked it well down the left before the latter picked out Molyneux who fired home from just outside the six-yard box. Biggins then came within a whisker of a quickfire second only to see his sidefooted attempt kiss the crossbar.
But Biggins made amends just after the restart, tapping home Joe Ironside's centre after he pounced on a defensive slip from Crewe's Mickey Demetriou.
And that's how it stayed although the visitors probed to try and extend their buffer ahead of Friday's return tie in South Yorkshire.
Here's how we rated the players.
1. Thimothee Lo-Tutala 7
Made some vital early saves and claims and dealt well with the aerial bombardment from the hosts. Photo: Andrew Roe/AHPIX LTD
2. Jamie Sterry 7
Up against pacy opposition down his flank, Sterry showed his experience throughout and also played some intricate forward passes. Photo: Howard Roe/AHPIX LTD
3. Tom Anderson 9
Produced a superb piece of defending with the score goalless and Crewe having a two-on-one counter. Aerially he bossed it all game. Colossus. Photo: Howard Roe/AHPIX LTD
4. Richard Wood 7
One dicey moment first half where he slipped and Crewe were in. Put in a much-needed block to deny Nevitt first half at 1-0 and was a rock second half. Photo: HOWARD ROE