Wood took four wickets in a fiery bowling performance to back up a half-century by England teammate Ben Stokes in Durham’s opening effort.
Durham now face Northants Steelbacks in the evening final provided the rain holds off.
Yorkshire had eight England players in their ranks at Edgbaston, but it was their international colleagues Stokes and Wood who stole the show.
Stokes, playing as a specialist batsman, underpinned Durham’s 156 for six with an eye-catching 56 from 36 balls that included three sixes.
And Wood then bowled with great pace and skill to finish with four for 24 as Yorkshirecame up short.
Durham started well after winning the toss and choosing to bat, but the introduction of spinner Azeem Rafiq in the fifth over tempted Keaton Jennings to go on the charge and Jonny Bairstow stumped him down the leg side for 11.
Liam Plunkett was the pick of the Yorkshire attack and he claimed his first scalp by bowling Surrey-bound Mark Stoneman for 25.
Bairstow missed another stumping chance, this time off Adil Rashid, with Michael Richardson on three and it proved costly as he and Stokes shared a stand of 72.
Stokes hit England colleague Rashid for six in an over costing 15 immediately after a brief rain delay, and international team-mate Joe Root was also dispatched into the stands.
Richardson eventually fell lbw for 29 playing across the line to Rashid, and Stokes was visibly annoyed after driving Rafiq to Tim Bresnan at cover shortly after bringing up his half-century.
Plunkett bowled Ryan Pringle to finish with two for 22 and Durham’s lower-order struggled to find momentum.
Yorkshire lost England pair Willey and Root cheaply as both openers perished with miscued pulls off Chris Rushworth.
Adam Lyth and Alex Lees looked to be making light work of the chase as 28 came from two overs bowled by spinners Scott Borthwick and Ryan Pringle, but Borthwick induced Lees to clip him to midwicket for 22.
Wood had delivered an excellent first two overs without luck, but the England seamer’s fortunes changed when a ripping yorker bowled Jonny Bairstow for three with the first ball of his new spell.
Gary Ballance fell two balls later for a duck, clipping a glance down the leg side only for Paul Collingwood to hold a tremendous catch at leg gully.
Lyth held firm at the other end to reach his fifty from 31 balls, but with 37 needed from 22 deliveries the left-hander carved Rushworth to Stoneman at backward point for 64.
Wicketkeeper Richardson dropped a routine edge from Bresnan’s first ball at the crease, although Rushworth still finished with outstanding figures of three for 19.
Leaning picked out Pringle at long-on from the second ball of Usman Arshad’s next over, but Bresnan hit the final ball for six over midwicket to leave 23 required from 12 balls.