Yorkshire had decided to bat first on a green surface but lost openers Lauren Winfield-Hill and Ami Campbell quickly to leave them 6-2, while star Australia batter Jess Jonassen only made nine, bowled by Charli Knott.
Sarah Glenn top scored with 30 and shared a partnership of 59 with Rebecca Duckworth (26), but they lost wickets at regular intervals with Charley Phillips claiming the final one to finish with 3-29 as the innings ended in the 43rd over.
Beth Langston took three quick wickets to leave The Blaze 51-3, but captain Marie Kelly played a fine anchor innings, with 66 off 87 balls to set up the victory.
There was a good fifth wicket stand of 51 with Prisha Thanawala (30) and although her and Kelly both fell with the score on 144, victory was sealed with the first ball of the 31st over.
The Blaze now have a 14-point cushion over fourth place and are looking a strong bet to make sure they are in the knockout stages in September.
After winning the toss, Surrey’s innings stuttered badly from 118-2 in the 21st over when a partnership of 56 between Ivreen Dhaliwal (22) and Alice-Davidson-Richards was ended with Dhaliwal falling to Abigail Glen.
From there, wickets fell at regular intervals as leg-spinner Katie Levick picked up an impressive 4-31 from eight overs.
Captain Davidson-Richards was the highest scoring batter with 26, although the highest contribution was the 33 in the extras column as Surrey were all out with 10.2 overs to spare.
When slow left-armer Bethan Miles took the wickets of Mady Villiers (21) and Emily Windsor for just four in quick succession as Durham slipped from 71-2 to 81-4, it was game on.
But Heather Graham joined Armitage at the crease to turn the run chase decisively in their favour.
Skipper Armitage struck seven fours in her 97-ball knock while Graham was unbeaten on 40 as they put on an unbroken 102-run partnership
Durham cruised home with 80 balls to spare for a fourth win in five in this competition, collecting five points as they jumped two ahead of Surrey in the process who suffered back-to-back losses.