Cook failed to survive the first over and Joe Root also fell cheaply before middle-order pair Gary Ballance (79) and Eoin Morgan (50) set the tourists on their way to 269 for seven.
Australia's response was ruthless, with an opening partnership of 163 between David Warner (65) and Aaron Finch, who went on to make 121 from 128 balls, helping lay the foundation for a six-wicket win.
And Cook admits his side will need partnerships like that if they are to banish memories of their Ashes whitewash and get something out of the five-match ODI series.
He told Sky Sports 2: "Hundreds win you games in one-day cricket and the last few games we haven't done that.
"We were a little bit slow to start off with. They bowled pretty well, give them credit for that, then we had the partnership between Gary and Eoin. But you've got to turn in big partnerships.
"You've got to get yourself in before you can do anything. The way Aaron and David Warner played, straight away they got themselves in, built a partnership and then accelerated along the way. In our innings, we kept building partnerships of 20 or 30 runs but then got out again, and you need to get double that to get a big score.
"We keep getting ourselves half in the game and not being able to cash in."
Cook also felt the bowling could have been a little tighter, but was prepared to allow excuses for an inexperienced attack including Boyd Rankin, Chris Jordan and all-rounder Ben Stokes.
He said: "A score of 270 would probably have been defendable if we'd bowled as well as we could have done.
"But we've a very inexperienced bowling attack, it was the first time in a big stadium for a lot of them, so you do have to give them a bit of leeway.
"I was very impressed with the way Jordan bowled. He attacked with the new ball, got some swing, and we could have had a breakthrough earlier.
"Still, we can turn this around. It's one-day cricket, you need people to come off on their day and we've got a lot of talent and a lot of firepower in that line-up to do that."
Australia captain Michael Clarke feels his team's current winning streak is a direct result of their indifferent form on last summer's tour of England.
"Our momentum has come from the series we played in the UK, when we lost the Ashes and turned that around," he said.
"Our one-day team is a different team to our Test team, but these guys are full of confidence. They're putting in the hard work and we've seen some results. They played really well in the UK and we've started this series on a good note as well."