The coach had to endure a poor first-half performance from his team, pre-match favourites England going into break all square with South Africa at six-all.
But with the prospect of an upset in the offing, England scored 19 unanswered points in the second half to ensure that they remain favourites to top the Pool – at the same time avoiding a quarter-final meeting with the All Blacks.
"It was a must-win game that we've won and we're very relieved," Woodward told ITV.
"It was a poor game but we've won it and we move on. There was a lot of tension going into the game. It was a big game, and one that we could have slipped up in, but we got through it.
"It was a hard contest and a pretty frustrating first half but we came through okay and now we can go to Melbourne with a smile on our faces."
Meanwhile, an equally relieved England skipper, Martin Johnson, praised fly-half Jonny Wilkinson's match-winning contribution.
"What can you say about him? He's fantastic – we're just happy he's on our side," said Johnson. "I'd rather we could dominate territory and score a few tries and kick conversions, but if it's not to be, it's not to be.
"His penalties were great, his tactical-kicking when we gave him more time was great, and his drop-goals kept us away from them in the last 10 minutes and stopped any chance of a comeback."