Coady, on-loan from Premier League giants Liverpool, scored in each half to move the Blades within seven points of sixth-placed Peterborough.
Three minutes before half-time and Crawley had a great opportunity to level the scores but Mark Connolly's shot in a crowded area sailed over the bar.
And the FA Cup semi-finalists doubled their lead in the 63rd minute following a blistering counter-attack.
Another excellent run down the left from Murphy saw the midfielder deliver a teasing ball and Coady, left unmarked at the back post, slotted home his second of the evening.
- Sports & Recreation
- Mark Connolly