Jonny Wilkinson booted his seventh kick from seven attempts eight minutes into injury time to secure the Lions a share of the spoils at the Millennium Stadium.
Woodward said: "We didn't expect to draw, we were hopeful to win the game but the biggest thing was to get through the game without any injuries.
"It was a difficult game. It was the first time the Lions have played a game like this before going on tour. I am just delighted no-one is injured and we can go on and prepare for the Bay of Plenty game.
"We have to move on and not over-react either way. It was a good start, we drew the game."
Argentina raced 13-0 ahead after just 15 minutes as Jose Nunez Piossek burst over for a try in the corner and the Pumas then held on as fly-half Federico Todeschini enjoyed a perfect evening with the boot.
Ollie Smith scampered over for the Lions but Todeschini's 20-point haul had seemed enough to keep the Lions caged before Wilkinson snatched the draw at the death.
"Jonny did well. He, like all the other players, found it difficult with the new combinations. The way he played and kicked speaks for itself," said Woodward.
"Michael Owen captained superbly, he made right calls at end. Under pressure he captained the team well."
Woodward added: "I didn't need this result to tell me, I'm very well aware how difficult this thing is. The Lions have been to New Zealand 10 times and lost nine.
"You can do lots in training but at least now we have got something to build on. We can go through the video and prepare for Bay of Plenty."