Barker took his match tally to nine wickets, on top of scoring a first innings century, with Warwickshire chasing down their target of 63 for victory inside 18 overs on the final afternoon of their LV= County Championship Division One clash.
Durham were still trailing, after being asked to follow-on on Tuesday, when play started on Wednesday with opener Mark Stoneman their main hope of avoiding defeat.
But he was helpless at the other end as Rikki Clarke trapped Paul Collingwood in front for 19 and Phil Mustard fell to Barker for eight to leave them in danger of an innings reverse.
Stoneman reached three figures off 179 balls but he fell just before Durham put their noses in front, disappointingly run out for 111.
Paul Coughlin resisted the Warwickshire attack on his way to 49, and there were also useful contributions from John Hastings (23) and Chris Rushworth (19), but when Barker (5-103) secured his five-fer, by dismissing the latter, Durham’s innings ended on 327, a lead of 62.
They briefly held out hopes of a very unlikely victory when Will Porterfield was caught off Rushworth’s bowling for two and Jonathan Trott was run out for six to leave Warwickshire on 12-2.
But Ian Westwood hit four fours in an unbeaten 32 and Laurie Evans added 13 in a 51-run partnership to see Warwickshire over the line with just over a session remaining.
The only other Division One match still in play, between Hampshire and Middlesex, ended in a predictable draw at the Ageas Bowl.
Hampshire batted on to 185-5 in their second innings before the captains shook hands with more than an hour remaining and no other result possible.
Michael Carbery scored a patient 57 for Hampshire, his innings coming off 130 balls, with the home side’s score taking up 77 of the final day’s overs as scoring proved tough.
All of Hampshire’s top six reached double figures, with Jimmy Adams contributing 37 and James Vince 32, but after tea Middlesex opted to use their occasional bowlers as the game wound down.
There was enough time for England opener Sam Robson to claim only his second first-class wicket as he had Adam Wheater caught behind for 13, the last wicket to fall before the draw was agreed.
Carberry earlier learned he was one of four players reprimanded by the ECB under their discipline code over an incident during the match against Yorkshire last week.