Cook's run included a profitable ninth-wicket partnership of 100 with Broad, whose own contribution of 56 runs made him the first England number 10 to post a half-century since 1991.
"It was amazing, very special to be out there for Cooky's 200. I don't think the Barmy Army stopped singing. It is awesome and helps as a batsman," Broad told BT Sport afterwards. "We had a bit of fun but I was careful not to get Chef too far out of his bubble. He has such a good zone and didn't want to do something too daft.
"Cooky said not to look too far ahead and just get five runs at a time - it's helpful for the lower order just breaking it down like that. I didn't want to take him out of his comfort zone, not trying to cause chaos and outscore each other.
"It was good fun and helpful to share the strike. When you get pressure on you as a lower order batsman having to face six balls from the same bowler, it's good to rotate the strike."
Cook will return to the crease on Thursday with James Anderson (0*).