Having lost the recent Test series 2-0, England were hoping for some respite in the 50-over format but after Friday's opener was washed out, the Proteas romped to an 80-run win in the second one-day international at the Ageas Bowl.
Amla's immaculately paced 124-ball innings pushed his side up to 287 for five, the fourth highest ODI score at the ground.
From the moment Lonwabo Tsotsobe bowled Cook with the second ball of the reply, the hosts did not look likely to chase it and they ended up all out 80 shy at 207 all out.
Victory saw South Africa replace England on top of the ICC rankings, making them world number ones in all three formats having already usurped their opponents in the Test arena.
Reflecting on a deflating performance, Cook said: "It was not one of our best days.
"We didn't play very well. South Africa did play well. Clearly when a guy gets 150 he has played outstandingly well and 287 was above par on that wicket. But we just never did what they did: build a partnership.
"One of the top three or four need to get a hundred. It's quite simple.
"We just didn't do that, for whatever reason."
Cook's one-day side have been on excellent form in 2012 and came into the match boasting a record of 10 wins and no defeats since the turn of the year.
They were well short of the form that saw them to series victories over Pakistan, West Indies and Australia in Southampton and Cook made no excuses.
"It's always frustrating when you don't play as well as you can, but sometimes that happens and there's no coming back from it in ODI cricket.
"It's not like a Test where you can drag it back, so it's been a frustrating day for us.
"You can name soft dismissals, we were sloppy in the field, we set very high standards with the ball and although we bowled well for some of it and for some of it we didn't.
"You can say all of those things. But if you're going to chase 280, someone in the top four or five needs to get a hundred because 20s and 40s aren't going to win you the game."
South Africa captain AB de Villiers was pleased to have reached number one in the ODI ladder earlier than expected but was quick to downplay the significance.
The mantle could be England's again by the end of the series, depending on results, and De Villiers is more interested in the tangible rewards of major trophies.
"I'm obviously proud of it (going to number one), it's a great achievement," he said.
"For the Test team it was probably one of our goals, with a very settled team for the last few years. But in the one-day and T20 sides there are a lot of new faces, a new captain and a lot of things still to achieve.
"We certainly weren't working towards getting number one in this series. We have a whole series to think about and the rankings can change every game.
"It's not something that is one of our main goals. We're looking way further ahead in the future - we have a World Cup in 2015, the Champions Trophy coming up.
"Number one is a bit of a bonus. We're very happy about it but it's not a main goal or a main concern at the moment."
England have the chance to square the series in Friday's third one-day international at the Kia Oval.
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