England started this morning needing six wickets to go 1-0 up in the four-Test series and they needed less than a session to do it as the world's top-ranked side were cast aside.
"It is really satisfying. It was just a good five days really - we played some good cricket and we were pretty relentless actually," Cook told ecb.co.uk.
"It is really pleasing to start the series well.
"You lose the toss and win by 250 you take that, don't you? I'd like to say I was going to bat first but that would be lying.
"We played really well from one to 11, there has been some stand-outs as well, everyone can feel in that dressing room really happy with the way they have gone about it."
Moeen Ali was named man of the match after he exploited a turning Kingsmead pitch to claim seven wickets in the Test while he crucially removed AB de Villiers with just the third ball of the day this morning.
"You can add James Taylor and Jonny Bairstow into that list as well," Cook said.
"It's a real collective effort from the lads and in one way me might have got a little rub of the green in this game that we didn't get in the UAE and that probably shows the fine margins that there are in international sport.
"We got those and we had a comfortable win. It was by no means easy, but really satisfying."
With just a three-day turnaround before the second Test in Cape Town, where South Africa have a proud record, Cook knows that his side can ill afford to rest on their laurels.
"Momentum is a funny thing because in our dressing room everything feels really good and I'm sure it doesn't feel so good in South Africa's dressing room at the moment," he said.
"It can change very quickly with one spell of bowling, one bad day from us and it changes. We understand that.
"We'll enjoy our first Test win away from home for quite a long time as a side, but fully aware that there is a real quick turnaround and how hard South Africa will come back."
If South Africa's impressive Test record at Newlands will offer them some comfort - they have only lost to Australia at the ground since 1962 - England have plenty of reason for optimism themselves.
England have not lost a Test series after going 1-0 up since 1999 - when they lost at home to New Zealand - but perhaps even more importantly they appear set to welcome back record wicket-taker James Anderson.
The 33-year-old was ruled out on the eve of the first Test with a calf injury and, while Cook was not ready to confirm his return, Anderson has come through his subsequent fitness tests.
"I genuinely don't know (if he will be fit to play). There are a lot of good signs though," Cook said.
"He's worked really hard with the medics over the last six days and he's coming through a lot of them unscathed, I think all of them unscathed so fingers crossed.
"We need to make that decision based on a number of things - five days in this heat is also quite hard to get through if you haven't bowled that much.
"It would be absolutely ideal if he could come back."