Goals from Henrik Larsson, Didier Agathe, Stan Petrov and Chris Sutton made sure there will be no shock exit from Europe for the Hoops.
Pre match talk of Hungarias being a physical side did not materialise, Celtic dominated the game from start to finish.
Henrik Larsson set a new record - he is now the top scorer in European competitions for any British side, with 31 goals. His delightful chip from 18 yards out on 18 minutes started off the scoring in a classy way.
Didier Agathe added to the Swede's opener on 36 minutes, and Celtic were cruising at half time.
Stan Petrov stayed cool just after the hour mark, to chip the ball past Hungarias' keeper, and Chris Sutton headed in a beautiful cross from Henrik Larsson after 91 minutes to all but secure Celtic's place in the Champions League.
Downpoints to the game were few and far between, the only real frustrating moment was Bobo Balde recieving a booking for blocking a player taking a shy - this could prove costly in a few games' time. Another was viewers watching on Setanta having to listen to Archie McPherson's rabble for 90 minutes!
Next week's return leg will be a case of taking care of business.
All in all, a good night for Celtic, it's now up to Martin O'Neill to ask the board to show him the money.
Celtic:
Douglas - 6
Valgaeren - 6
Balde - 7
Varga - 7
Lennon - 7
Lambert - 7
Petrov - 9
Thompson - 7
Agathe - 6
Larsson - 8
Sutton - 8
Celtic-Mad Man of the Match: Stan Petrov