Supported by 89 from Steven Mullaney, Root hit two sixes and 10 fours in his 93-ball first List A century, as Notts recovered from 92-4 to reach 303-6.
Jonathan Trott hit 14 fours in his 104, his 21st limited-overs century.
But the holders fell short on 293-9 to lose by 10 runs, a third straight loss.
Warwickshire needed 49 from the last five overs as the pressure mounted, and after Rikki Clarke and Keith Barker both holed out, that became 16 needed off the last over, bowled by Luke Fletcher, but two run-outs sealed the hosts' fate.
Bears captain Ian Bell made 51, but he was the second of three victims for Mullaney, who crucially accounted for Trott, no sooner than the former England batsman had reached three figures - for the fourth time in his last 10 One-Day Cup innings.
After starting their defence of the trophy with an impressive 10-wicket win against Warwickshire, the Bears' third straight defeat now leaves them second bottom in North Group ahead of Sunday's visit of winless Lancashire.
Notts climb to fourth in the group, three points behind leaders Worcestershire, ahead of Sunday's home fixture against Leicestershire at Welbeck Colliery.
Warwickshire's Jonathan Trott told BBC WM:
"We lost our way a little bit with the bat and lost too many wickets at crucial times, including myself. I shouldn't have got out when I did.
"If I had got a few more runs, 120 or 130, it would have been a lot easier for the rest of the guys coming in.
"Mullaney and Root batted well to get them up to a defendable total but the disappointing thing is that we had chances in the game to close it out and win it and unfortunately we couldn't do it."
Nottinghamshire batsman Billy Root told BBC Radio Nottingham:
"The century just snuck up on me really. I wasn't thinking about it. It was just a question of trying to score as many runs as possible.
"There were a lot of high-fives in the dressing-room which was nice. I haven't checked my messages yet but I am sure there will be one from Joe.
"It's nice to be selected for the start of a competition and then it's your job to try to contribute and I am happy to have done so. It's a great dressing-room to be in at the moment. Everyone is in such a good place."
Nottinghamshire's Billy Root, younger brother of England Test captain Joe, hit an unbeaten 107 at Edgbaston to help them beat Warwickshire and record a second successive One-Day Cup win.