Joe Root said he was "proud" to set a new record for the number of Test wins as England captain, and paid tribute to the players that helped him do it.
"You don't do that on your own as a captain," said Root.
"You're the one making the decisions, but they are the ones putting in big performances."
Root, 30, took charge in 2017 and has led England in 55 Tests.
His mentor Vaughan captained in 51 matches between 2003 and 2008, though suffered only 11 losses, to Root's 20.
The Yorkshire duo also both played their club cricket at Sheffield Collegiate.
"I'm proud to have gone past Michael," said Root.
"I'm living my boyhood dream. Captaining England is something I dreamt of doing when I was really small."
Whereas Vaughan famously led England to their 2005 Ashes success, Root's two series against Australia have resulted in a 4-0 away defeat in 2017-18 and a 2-2 draw at home in 2019.
He is set to lead England in their bid to regain the urn down under this winter.
"Joe is tough," Vaughan told BBC Sport. "He might have that lovely smile, but inside there's a lot of steel.
"The only feather that is left for Joe to put in his cap is that little urn.
"I've said from day one, he's done everything the same as me. Same club, same county at Yorkshire. He's nicked everything from me bar my wife!"
Root is four matches behind Alastair Cook's record of leading England in 59 Tests and said he would "not put a time limit" on how long he will stay in the job.
"As long as I'm enjoying it, as long as I feel like the right person to take this team forward, that will be my focus," he said.
"As long as I feel like I'm the right man for it in my own mind, I'm more than happy to keep doing what we are."
England's huge win at Headingley, wrapped up before lunch on the fourth day, came after they suffered a devastating final-day loss in the second Test at Lord's.
They have levelled the five-match series at 1-1, with the fourth Test at The Oval beginning on Thursday.
"It was a very clinical performance," said Root. "We're sat here now at 1-1, with a big smiley dressing room and two big games ahead of us.
