Following a first half of few chances, Alex centre-back Dugdale headed home Ashley Westwood's 49th-minute free-kick to end the visitors' run of five consecutive clean sheets and send Gresty Road into raptures.
It was no less than Steve Davis' side deserved as they extended their unbeaten run to 17 matches by outplaying a Southend team who recorded a league double over them this season.
Paul Sturrock's men also headed into the game in a rich vein of form, winning four of their last five to miss out on automatic promotion by a point, but turned in a subdued performance.
Michael Timlin, Bilel Mohsni and Ryan Hall all went close as a belated rally arrived from Southend, but they emerged with nothing to take back to Roots Hall for Wednesday night's second leg.
Crewe skipper David Artell was a surprise absentee due to a broken bone in his foot, while right-back Carl Martin returned from a 12-game absence to leave Billy Bodin among the substitutes and Matt Tootle deployed in an unfamiliar left-midfield role.
Kane Ferdinand replaced Freddy Eastwood in Southend's only change.
Tootle made the most of his attacking license in the opening exchanges and almost got on the end of Byron Moore's dangerous eighth minute cross, Southend full-back Sean Clohessy on hand to clear behind.
Ajay Leitch-Smith was narrowly beaten to Tootle's through-ball by U's goalkeeper Cameron Belford before strike partner Nick Powell was crowded out following some clever footwork in the area.
In the 23rd minute Belford did well to punch Powell's left-wing centre to safety as Luke Murphy's late run left him momentarily unsighted.
Southend edged their way into the contest after the half-hour and Ferdinand headed over from Hall's 41st-minute cross, but half-time arrived without Belford or opposite number Steve Phillips being overly troubled.
Hall had a shot blocked following a deft outside-of-the-foot pass from Mohsni as Sturrock's men began the second period with greater intent, only to fall behind.
There appeared to be little danger when Westwood launched a speculative ball into the United area, but Dugdale rose majestically to send a looping header beyond Belford into the far corner.
Crewe sought to press home their advantage with the visitors wobbling, and Tootle just failed to find the decisive touch from another raking Westwood delivery.
Belford produced a phenomenal save to push a Leitch-Smith effort over the bar in the 66th minute after a long clearance split the Southend defence.
Southend finally enjoyed a clear opportunity with 20 minutes remaining when Mohsni cushioned a header to the onrushing Michael Timlin, but Phillips parried well.
Mohsni - who scored three of his 13 league goals this season against Alex - then nodded over at the back post before Hall fizzed a 20-yard drive wide and Crewe held on.