Four wickets each from Stuart Broad and Jack Leach did for the visitors, who will be content at retaining the urn.
Root was thrilled by the response to defeat at Old Trafford in the fourth Test and singled out Jofra Archer for special praise.
Root said: "We were excellent this week. Having lost the toss, to play as we did was fantastic. We drove the game all the time and got there in the end.
"Jofra makes things happen, and he's backed up with other skilful bowlers. Collectively we've shown a lot of character throughout the whole series.
"We were not always at our best, but we've fought really hard. Credit to Tim (Paine) and his team."
England's man of the series Ben Stokes reiterated his disappointment at not winning the Ashes, but admitted the drawn series ensured some pride.
Stokes said: "Obviously it was disappointing in Manchester for us to know that we couldn't get the Ashes back.
"But we turned up here this week knowing that we still had a huge amount of pride and the Test Championship to play for, and we managed to draw the series, which was the goal at the end of this game."
While man of the series Steve Smith paid tribute to a "spectacular" contest.
Smith said: "It's been an amazing couple of months in England. The cricket's been spectacular. The series has ebbed and flowed.
"I've loved every minute and am proud to have been able to perform for Australia and bring the urn home.
"The middle of the wicket still played pretty well, and Wade applied himself, batted beautifully today. But England played terrific cricket."
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