The captain said he was proud that England held on for a draw in the fourth Test to avoid a whitewash.
After being set an unlikely 388 to win, England's last man James Anderson hit the final over to leave them 270-9.
"To do that was really nice," said Root.
There are a lot of improvements we can make, but it's a step in the right direction.
The series was already lost by England when they lost the third Test in Melbourne.
The tourists were beaten for the third time in 15 years and England lost for the last time.
Chris Silverwood missed the match after a number of Covid cases in the backroom staff.
Bairstow and Buttler were both injured and will miss the final Test because of a broken finger.
"Relief is the most overwhelming emotion," said Root.
Guys have stood up and showed a lot of character, showed a lot of pride, and showed how much they care about playing for this team.
Silverwood will be in charge of the fifth Test in Hobart, which starts on Friday.
Bairstow's thumb injury and the side strain of Ben Stokes will be assessed by England.
England began the final day in Sydney at 30 0 and all three injured players were batting.
Buttler used up 38 balls for his 11 while Bairstow used up 41.
Stuart Broad defended an over from Nathan Lyon before number 11 Anderson resisted Steve Smith after Jack Leach's leg-spin slipped with 12 balls remaining.
"We don't make it easy for ourselves, do we?" asked Root.
I spoke before the match about putting some pride back into English cricket. The fight, desire and character shown in the five days has done a small amount.
The last Test was a dark day. It would have been easy to roll over for the last two.
It's important that you front up when you're out of the series. Everyone has done that this week.
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