Celtic, the holders, with only Bobo Balde and Stilian Petrov of their regular line-up included, dictated the pace more or less throughout, although early on, at least, the second division side made it an old-fashioned cup-tie in every sense.
Mark Cowan sent a good header from a Jimmy Fisher free-kick just wide after five minutes, Gregg Watson was booked for a foul on Shaun Maloney and Alloa manager Terry Christie, a headmaster, was given a taste of his own medicine when he was sternly lectured for some touchline antics by the referee.
That initial flurry over, however, it was on to what was expected and after 19 minutes, and following Derek Soutar's fine save from a Petrov free-kick, Celtic took the lead.
There seemed little danger when former Toulouse player Balde moved forward, but the big defender unleashed a speculative 25-yard drive that took a wicked bounce off the less-than-perfect pitch and although Soutar did get a hand to the ball, it still finished in the net.
The rest of the first half was all about Celtic trying to add to their lead and goalkeeper Soutar excelled with fine saves to deny Balde, Bobby Petta, Olivier Tebily and Momo Sylla. Ian Little was then booked for a foul on Balde as he also tried to stem the tide. Three minutes from the interval, though, it was 2-0 as Morten Wieghorst celebrated his return after illness when he headed a Jackie McNamara cross past Soutar.
The second half was never likely to be much of a contest and Celtic, with Jamie Smith on for Petrov, went three up when the substitute combined with Sylla to give Maloney a chance he accepted gratefully in the 56th minute.
Celtic were in total control and Petta made it 4-0 with a fine solo effort after 75 minutes. Sylla then scored the fifth - his first for the club - from close range seven minutes from the end. The victory sets up an all-Premier League fourth-round tie at Kilmarnock.
Meanwhile, the Dundee United chairman Doug Smith and the board have been overthrown by Jim McLean's consortium after shareholders voted in favour of the move at yesterday's e.g.m.
With McLean, the club's former manager and chairman, holding 42 per cent of the shares and the consortium 54 per cent, the result came as no surprise. Smith said: "For this to happen is despicable and not in the best interests of Dundee United."
ALLOA: Soutar, Valentine, Watson (Gareth Evans 60), Thomson, Cowan, Hamilton, Christie, Fisher, Whalen (Walker 57), Hutchison (Curran 73), Little. Subs Not Used: Knox, Jim Evans. Booked: Watson, Little.
CELTIC: Gould, Boyd, Balde (Healy 61), Tebily, Guppy, McNamara, Wieghorst, Petrov (Smith 45), Petta, Maloney, Sylla. Subs Not Used: Lambert, Moravcik, Douglas. Goals: Balde 19, Wieghorst 42, Maloney 56, Petta 75, Sylla 83.
Referee: M McCurry (Scotland).
