Wilkinson moves back to number 10 from inside centre, with his selection coming at the expense of a dropped Stephen Jones and an in-form Charlie Hodgson.
Hodgson still cannot find a place in the Lions' Test 22, despite some outstanding performances on tour, notably against Taranaki and Otago.
But Woodward has no doubt about Wilkinson's ability to shine amid the pressure of a game the Lions must win, otherwise their Test series hopes will vanish.
"If you actually watch what Jonny did last Saturday, he was probably the best back on the field in terms of the Lions," said Woodward.
"He is mentally and physically up to it - I think he is the big-match player. I think Jonny is the man, he has the goalkicking capacity and he is the best goalkicker.
"It is a tough call, and I can understand people would make certain assumptions about me picking him, but I believe in the cold light of day, he is the right guy to play there.
"It is very competitive, but I do believe Wilkinson is the number one fly-half in the party."
Woodward though, hinted that he might have stuck with the Jones/Wilkinson combination at 10 and 12, had captain Brian O'Driscoll not suffered a tour-ending shoulder injury after less than two minutes of the first Test.
"To be fair, if O'Driscoll had not got injured, I was very excited about playing Jones and Wilkinson together," he added.
"We could never see that development last Saturday, because we just got wiped out in the lineout.
"I don't think whoever played in the back division last Saturday would have made any difference. When you get taken to pieces at the lineout and you lose most of your own throws, it's difficult.
"Once O'Driscoll was out, it didn't look right anymore playing Jones and Wilkinson together. It is obviously very tough on Stephen Jones, but it is also tough on Charlie Hodgson and Ronan O'Gara, who have also been having fantastic tours."
Woodward is maintaining his unshakeable belief that his Lions can square the series at Westpac Stadium.
"It is a game we have to win, and I have put out a very attacking side," he said.
"I think when you play so poorly, you have to do something. I still sit in my room and look at that (first) Test team I picked, and in my head, it still makes total sense in terms of the type of game we were going to try and play.
"But you have to act decisively and make sure you are fair to everyone in the party, and the real form players like Simon Easterby and Ryan Jones deserve their chance.
"Once we got beaten so badly, I had to change things. I have always believed in everything I have done, that if you make a mistake, you fix it quickly. You don't dwell on it and try and justify it."
And as for the post-first Test criticism that has been levelled at him, Woodward added: "Sometimes, I am a little bit amazed at the amount vitriol and bile that comes your way, and you don't forget what certain people say, especially when you are down.
"But it is all part of the job. I have been here before, and I guess I will be here again at some stage in the future, but it does surprise me sometimes the reaction of one or two people when you are down."