League One Coventry sit 64 places above Worcester in the English football pyramid, but they were undone by their non-league opposition, who made the most of having a numerical advantage for 51 minutes.
Coventry goalkeeper Lee Burge was dismissed six minutes before the break following an altercation with Daniel Nti and Sean Geddes converted the resulting spot-kick to give the visitors the lead.
The hosts were given a lifeline when James Maddison won a debatable penalty, but Reda Johnson saw his effort saved by Nathan Vaughan and Worcester capitalised just after the break.
Geddes found the net for a second time with a fine low drive from just inside the area and, although Johnson made amends for his earlier miss nine minutes from time, Worcester held on to send the 1987 cup winners out of the Cup.
Portsmouth, winners in 2008 and finalists in 2010, were held to a 2-2 home draw by Conference National side Aldershot.
The hosts took the lead through Jed Wallace's 16th-minute penalty, but they were pegged back by goals from Jordan Roberts and Mark Molesley.
Danny Hollands ultimately spared Portsmouth's blushes in the 81st minute and, although Paul Robinson was dismissed a few moments later, the hosts held on to force a replay.
The highest scoring contest of the round saw Colchester United defeat hosts Gosport Borough 6-3, while Chesterfield also hit six on the road, beating Braintree Town 6-0.
Norton United were the lowest ranked team in action on Sunday and they were ultimately thumped 4-0 by Rob Ramshaw-inspired Gateshead, with the forward scoring three times.
Bradford City came from behind to beat Halifax Town 2-1 in the day's early match, sparing themselves from an embarrassing exit against their Yorkshire rivals, while Altrincham lost 4-1 to Blyth Spartans, and Wrexham beat Woking 3-0.
Scunthorpe United won 2-0 at Forest Green Rovers, while Maidstone United and Accrington Stanley held Stevenage and Notts County respectively in the only goalless games of the day.
