Moeen, who was named Man of the Match, took four wickets in the second innings, including the key scalp of Virat Kohli to break a threatening century stand with Ajinkya Rahane, to follow his five-wicket haul on the second day which kept England in the match.
It was Moeen's first Test since he was dropped in New Zealand in March and he also contributed an important 40 to halt England's first innings collapse, then was promoted to No. 3 in the second innings.
"I think today was the best I've ever seen him bowl in an England shirt, that's such a testament of how he's gone about things and how he's come back, his strength of character and that strength in depth we have as a squad," Root said. "A bit of time away from this environment, I've spoken to him about that - having a different look at things, going to IPL and coming back performing really well for Worcester must give you a huge amount of confidence."
As Kohli and Rahane forged their stand during the afternoon session - Kohli surviving a very close lbw review when Joel Wilson, the TV umpire, ruled he had got an inside edge - it appeared India were moving into the ascendancy, but Root always retained faith they could break through.
"We could be quite defensive with the field, keep the catchers in and trust there'd be one ball in there to turn the game around," he said. "More than anything it was about keeping the guys calm, which wasn't too difficult as captain. The way we went about and trusted our process, how we wanted to approach the day was exceptional. If we can take that forward it will stand us in good stead."