He stood one under with five to play, while big-hitting Quiros, hoping to mark the centenary of the championship with another home victory, had to settle for a one over 73.
“I’m very happy, I played really solid,” said Molinari. “It was as tough as it can get.
“The rough is just brutal, and the wind and rain make it hard to keep the ball on the fairway, but I played very well.
“It’s a shame that I had two birdie chances on the last two holes and I made a bogey and a par, but overall I am very happy. The greens are quite soft, but it's really hard to get the ball on the fairways.
“This course should favour straight players like me, but when you show up on the first tee you never know what’s going to happen. My game was really good today, and hopefully it will be the same for the next three days and I’m sure I will have a chance to win this tournament on Sunday.”
Wood carded an eight on the 514 yard fifth, but finished with a birdie to post a level par 72 that was good enough for a share of 12th place.
Siem bogeyed the last for a 71, but Molinari still had company at the top - Hebert remained two under with three to go while Spaniard Alejandro Cañizares birdied two of the first four holes to make it a three-way tie.
Molinari was relegated to third place when 19 year old compatriot Matteo Manassero had an eagle, two birdies and a bogey in the first six holes and England's Sam Hutsby birdied three of the first six holes on the back nine.