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.
Ashes-holders England came into this match having already retained the urn after a rain-affected draw in the third Test at Old Trafford left them 2-0 up with two to play.
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.
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.
England, on a slowish outfield, initially found runs hard to come by against Ryan Harris and Tasmania’s Bird, playing his third Test.
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.
Nevertheless, the much-criticised Hot Spot — whose inventor Warren Brennan suggested ahead of this match was being deliberately duped by players on both sides applying silicone tape to their bats — thermal imaging element of the controversial Decision Review System showed a mark and England were 34 for one.
England reached lunch on 57 for one, with Cook 21 not out and Trott unbeaten on 13.
Like Cook, Trott had yet to make a major score this series but he looked in good touch as he cut Peter Siddle for four and swept Lyon for a boundary.
But, trying to whip Lyon through his favourite onside region, Trott was well caught off bat and pad by diving short leg Usman Khawaja.
Together with Cook, he’d put on 73 for the second wicket.
Pietersen, as had been the case when he made a century at Old Trafford, was determined Lyon would not dictate terms.
He drove a Lyon half-volley down the ground and next ball went down the pitch to loft him over mid-off for another four.
Cook, who would have been run out by yards on 41 if David Warner’s throw had hit the stumps, went on to make a 153-ball fifty when he edged Bird for the fifth four of his innings.
Pietersen then fell tamely when, with Lyon bowling around the wicket, he opened the face and got a thin edge to wicketkeeper Brad Haddin on 26.
Angered by suggestions in a report by Australia’s Channel Nine television that he was one of the players taping his bat to trick Hot Spot, Pietersen walked off without waiting for Hill’s decision.
And 149 for three became 153 for four when Bird had Cook lbw.