Dugdale’s ninth-minute header handed the Railwaymen an early boost as they knew victory would keep them abreast of the bottom four against a Preston side already assured of a play-off place.
The centre-half met Jon Guthrie’s corner with a towering far-post header, which flew into the top corner.
Uche Ikpeazu went close to doubling Crewe’s advantage with the Watford loanee out-muscling Bailey Wright only to drag his shot across the face of goal.
Preston controlled much of the possession, but struggled to make a breakthrough in the first-half.
Neil Kilkenny drove inches over from 25 yards and Joe Garner’s effort looked to be veering past goalkeeper Ben Garratt, but the youngster stuck out a glove to turn the ball around at the start of the second period.
But the hard work of Ikpeazu helped Crewe into the comfort zone on the hour-mark with the striker putting Scott Laird under pressure on the by-line and pulling the ball back for Pogba to side-foot a second.
Pogba could have put the three points in the bag, but failed to connect cleanly with Byron Moore’s cross, miscuing his header.
Substitute Chuks Aneke forced Preston goalkeeper Declan Rudd to tip his chip on to the crossbar as Crewe stayed up in style before John Welsh reduced the arrears for the visitors with a left-footed shot from the edge of the penalty area in stoppage time.