The 18-year-old fired the home side in front after 41 minutes when he skipped past Rovers defender Ash Taylor and smashed the ball into the top corner from inside the box.
And in the 83rd minute Conclough doubled his and Crewe's total for the afternoon when he received Byron Moore's cross on the six-yard line and converted an easy finish.
The damage had already been done for the visitors when Danny Holmes was shown a red card for a second bookable offence in the 33rd minute.
The right-back was cautioned early on for a foul on Moore before a rash tackle on the same man earned him a red card.
Ronnie Moore's Tranmere did pull one goal back after 87 minutes when Jean-Louis Akpa Akpro tapped home Andrai Jones' cross.
But it proved to be nothing more than a consolation for the visitors, who remain without a win in the league this season.