Ajay Leitch-Smith opened the scoring at Gresty Road and Max Clayton also bagged a brace as the Railwaymen made it a club-record 20 games unbeaten under manager Steve Davis.
A goal in the seventh minute helped Crewe settle against their npower League One counterparts. Ashley Westwood's free-kick was blocked but Leitch-Smith did not hesitate in lifting the loose ball over goalkeeper Scott Flinders and into the roof of the net.
Crewe doubled their lead with 12 minutes gone when, after Leitch-Smith was thwarted by Flinders, Pogba picked up the loose ball and squared for Clayton to drill into the bottom corner.
Pools striker Steve Howard headed over the crossbar and saw another close-range effort superbly saved by Alan Martin before the Alex again showed their superior killer instinct. Byron Moore crossed for Pogba to sidefoot home in the 34th minute before sending clear Clayton, who calmly slotted past Flinders.
Combination play between Leitch-Smith and Moore gave Clayton the chance to complete his hat-trick, but the teenager's rising volley flew inches over the bar. Pogba netted his second in injury time, shrugging off two defenders and stabbing home.