Chasing 271, the visitors looked well set on 151-1 in the 29th over before Daryl Mitchell (3-38) and John Hastings (3-50) tore through the Durham order.
Graham Clark struck 114 and Keaton Jennings 47, but no other visiting batsman reached 20 as they made 255-9.
The hosts, who guaranteed themselves a quarter-final place, made 270-8 led by Brett D'Oliveira (73 not out).
Opener Tom Kohler-Cadmore compiled 48 from 59 deliveries, while England all-rounder Moeen Ali struck six fours in his 34.
The hosts still looked to be falling short when Durham's opening pair shared a partnership of 122 before Moeen took the key wicket of Jennings.
Worcestershire, who have now won five of their seven One-Day Cup matches to overtake Yorkshire, will finish top of the group and secure a home semi-final if they win their day-nighter on Tuesday at Derby.
Durham are fourth after four wins and two losses, but their hopes of a quarter-final place no longer rests in their own hands.
They need to beat Lancashire on Tuesday and hope Notts slip up against Northants.
Worcestershire skipper Joe Leach told BBC Hereford & Worcester:
"We probably looked a bit dead and buried after 25 overs but it is testament to the belief in the side and the run we are on that we believed we could win.
"That showed and as soon as we got that first wicket, we swarmed all over them and they panicked a little bit. In days gone by we might have folded and got hammered but we've got a very talented group.
"To show that character stands us in very good stead, especially going into what are the knockout stages of the competition now."
Durham head coach Jon Lewis told BBC Newcastle:
"I don't think it ever looked quite like a stroll. It was a good partnership. But it never quite got the run rate down. It held the run rate steady.
"As a fielding team, you know if it is still staying steady and you get a new guy at the crease, you can try and impose yourself which, to be fair to Worcestershire, they did very well. They squeezed the new batsmen very well and we stopped dictating the terms.
"It was great for Graham to get his first century but I'm sure even for him it will be a little bit bitter that it is not in a winning cause."
Durham lost eight wickets for 87 runs on their way to a 15-run defeat at new North Group leaders Worcestershire.