Owen's goal saved the visitors' blushes after they threw away a 2-0 lead following an own goal by Scott Dann and a second from James Milner.
Clinton Morrison had pulled one back for City in first-half injury-time before Dann's last-gasp leveller forced extra time.
Both sides started well but it was Newcastle who made the breakthrough in the 22nd minute when Milner's low cross caused a mix-up between Dann and goalkeeper Andy Marshall, which ended up with the ball bouncing off the defender's leg and into the net.
Kevin Keegan's men pushed and came close twice through Charles N'Zogbia.
Their pressure was rewarded in the 33rd minute when Milner's hopeful effort from 25 yards clipped Guillaume Beuzelin's foot and sailed into the net.
But just as Newcastle seemed to be cruising, Morrison hit an injury-time screamer.
His effort from 20 yards flew past Shay Given into the top corner to leave the visitors stunned.
The second half produced much of the same and efforts from Milner and Danny Guthrie nearly extended the lead.
Owen was introduced after 75 minutes but had little to do as his side protected their advantage.
But they were shocked again in stoppage time when Aron Gunnarsson's long throw into a crowded area was finished by the head of Dann.
In extra time it was the fresh legs of Owen which made the difference. He beat City's offside trap in the 97th minute and chipped the ball past Marshall to restore the lead.
Coventry nearly levelled again when Elliott Ward's header thundered against the crossbar in the 115th minute, but Keegan's men held on for the victory.