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Counago's touch set up the midfielder and he slid in his first goal for the club from 10 yards out.
The early goal settled any nerves Ipswich may have had and on 18 minutes it was 2-0, Bent fed Counago, who side-footed into the corner of the net.
Only another two minutes had passed before Counago and Miller each scored again.
First Counago centred from the right for Miller who, with his back to goal, flicked a back heel into the far corner.
Then Miller returned the compliment, crossing from the left for Counago to turn in the fourth goal.
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The second half was a mere formality although it was the visitors who scored next, Gregory Molitor on target 11 minutes after the break.
John McGreal scored Ipswich's sixth on the hour, tucking the ball away after Fabien Wilnis's shot had been blocked by goalkeeper Hartert.
Counago completed his hat-trick on 73 minutes and substitute Darren Ambrose completed the rout when he turned home a cross from fellow sub Finidi George 11 minutes from time.
Ipswich: Marshall, Wilnis, McGreal, Brown, Clapham, Tommy Miller, Magilton, Holland, Reuser, Counago, Marcus Bent. Subs: Pullen, Ambrose, Darren Bent, George, Gaardsoe, Richards, Naylor.
Beggen: Hartert, Calvaruso, Laraccia, Ney, Miedico, Molitor, Lionel Da Silva, Gomes de Almeida, Martin, Da Luz, Pace. Subs: Gilbertz, Picard, Lopes, Hamdaoui, Tom Schaack, Diallo, Schmit.