Woodgate shone on his debut after his £7m move from Middlesbrough as defences dominated.
Everton manager David Moyes said afterwards: "I'm pleased with a point. We are little bit short of players at the moment, but I was really pleased with the effort and it was a great point.
"I thought there was a bit of tiredness near the end, but we have been on a good run. We've had the African Cup of Nations boys away, but we have won two league games and drawn one.
"Hopefully we will get them back soon and kick on from there."
Spurs assistant coach Gus Poyet admitted: "I thought Jonathan Woodgate was absolutely fantastic. He was commanding the back four and winning everything in the air.
"He was tight when he needed to be and made great decisions. Sometimes the good players are the ones that make the best decisions and Woody does that. We were very pleased with him."