England, who won the toss, were 155 for four at tea with Ian Bell six not out and Jonathan Bairstow nought not out after off-spinner Nathan Lyon had taken two wickets for 12 runs in 16 balls to dismiss South Africa-born duo Jonathan Trott (49) and Kevin Pietersen (26) and Bird snared Cook for 51.
Opening batsman Cook had been at the crease for nearly four hours when Bird, in for the dropped Mitchell Starc, deceived the left-hander into leaving a ball that swung in the air and cut in off the pitch to have him lbw.
But Australia could still deny them a series victory by winning both this match and the fifth Test at The Oval.
Cook opted to bat despite the pitch and overhead conditions promising assistance to Australia's seamers in the first Ashes match at the headquarters ground of northeast county Durham.
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His decision also meant Cook chose to face the new ball first-up even though his own form this series -- 145 runs in three Tests at 24.16 with two fifties and a best of 62 -- had been modest.
However, it was first change Shane Watson who had Joe Root (16) caught behind, although Australia had to challenge New Zealand umpire Tony Hill's original not-out verdict.