Swiss international right-back Haas fired Albion ahead with a sixth minute volley - his second Carling Cup goal of the season.
Substitute Dobie then booked Albion's place in the last eight with his first touch of the ball after 56 minutes.
Albion could even afford the luxury of seeing midfielder Jason Koumas miss a first-half penalty as the First Division leaders swept aside the United youngsters.
United only included one player - John O'Shea - who had started Sunday's 1-0 defeat at Chelsea.
Danny Pugh, Phil Bardsley and Paul Tierney all made their debuts for United.
United struggled to make much of an impact and when they did they were frustrated by Albion goalkeeper Russell Hoult.
Haas left United reeling when he finished off a fine four-man move with a spectacular volley.
Koumas picked out wing-back Neil Clement on the left flank and his cross was headed into the path of Haas by striker Rob Hulse.
Haas let fly from the edge of the area with a drive that flew past the stranded United keeper Roy Carroll.
United's youngsters looked overawed in the opening stages and Albion should really have taken advantage of that only for Koumas to fluff his penalty.
Koumas had won the spot kick himself when he was sent tumbling by Pugh just inside the area.
He tried to stroke the spot-kick past Carroll but the Northern Ireland international was able to dive low to his left to make a smart save.
Having been forced to overcome a rocky start, United began to work their way into the game.
It was no surprise that their one "old" head, Nicky Butt, led the bid to force the team back onto level terms.
Butt had two chances in the final fives minutes of the first half only to be denied by Albion keeper Hoult.
Albion's senior players always carried a greater threat in attack and only the woodwork prevented Danny Dichio heading them further in front after 48 minutes.
Kleberson briefly raised United's hopes when he saw his drive beaten away by Hoult 60 seconds later but their fate was then sealed by Dobie.
The Scottish international striker had been on the pitch less than 60 seconds when he stooped low to steer a cross from Clement beyond Carroll.
Hoult then denied Kleberson for a second time after 72 minutes with another flying save as United's Carling Cup campaign came to a dismal end.