The lively Rodgers opened the scoring in the 28th minute, calmly placing the ball past Crewe goalkeeper David Button for his fifth goal of the season, before Westcarr added the Magpies' second after the interval when he found the bottom corner from distance.
To add insult to injury, the Railwaymen even saw Kasper Schmeichel save Steven Schumacher's penalty after John Thompson had fouled Joel Grant.
The alarm bells had been ringing for Crewe's defence, who have now shipped 19 goals in their last seven games, from the first whistle.
Westcarr hit the crossbar after a goalmouth scramble early on, while Rodgers' looping cross almost deceived Button.
Crewe had chances of their own, but could not find a way past Schmeichel. Grant swivelled past Mike Edwards to get in on goal, but saw his effort saved one-handed, while Shaun Miller's late stab at goal was clawed away by the Dane.