Chasing 221 to win, the visitors were 140 for eight in the 42nd over when Ajmal Shahzad joined Tim Bresnan.
But the target was down to six off two balls when Bresnan was last out for 55, driving a slower ball from Graham Onions to Dale Benkenstein at extra cover.
The target still looked way beyond Yorkshire with 63 needed off six overs, but it was down to 44 off four when Neil Killeen returned.
His first seven overs with the new ball had seen him take 2-11, but both Bresnan and Shahzad drove him for six as 22 came off the 47th over.
With 15 needed, Shahzad drove the first ball of Killeen’s next over into the hands of Liam Plunkett at long-off and the target came down to 11 off the last over, but Onions was equal to the task.
A day after his arrival in the north-east, South African Neil McKenzie held Durham’s innings together by making 77 out of a total of 220.
Killeen then took the first two wickets before making way for Steve Harmison with Yorkshire on 49 for three after 17 overs.
Although he occasionally strayed in line, Harmison cranked his pace up to almost 90mph and struck with the first ball of both his spells.
Gerard Brophy tried to force him through the off side off the back foot and played on, then Richard Pyrah holed out to Steve’s brother Ben at deep midwicket.
The expected clash between Harmison and Matthew Hoggard in their battle to get back in the England team did not materialise.
Hoggard joined back injury victim Darren Gough on the sidelines because of a stiff neck, while paceman Morne Morkel has been delayed for 10 days by South African domestic Twenty20 commitments.
Durham would be grateful that there was no such delay for McKenzie, who will be with them for the first seven weeks and will bat at four after opening the innings with great success in South Africa’s last two Test series. Neil was also named as the Friends Provident Trophy Man of the Match on his debut.
The value of his 123-ball stay, with five fours and two sixes, was underlined by the second highest score being the 25 off 22 balls with which Phil Mustard launched the innings after Durham were put in by Anthony McGrath.
The England one-day wicketkeeper also pulled off two stumpings as Yorkshire desperately tried to break free of the shackles imposed by the superior Durham attack.
On the slow pitch McGrath probably felt he had made up for the absence of his big guns by squeezing 17 overs of tidy medium pace out of himself and Pyrah.
But the pick of his bowlers was Shahzad, who reached 85mph and was unlucky that the one he nipped back sharply to bowl Kyle Coetzer provided his only wicket.
Bresnan finished with 3-51 when he took the last wicket by bowling Harmison with the first ball of the 50th over.
Click here to see the scorecard – www.ecb.co.uk/stats/fixtures/durham-v-yorkshire-20-4-08,13881,FF.html