The two National League clubs are creating history but becoming the first English clubs to play in a Scottish domestic competition.
They earned the right by being the highest finishers who didn’t earn promotion last season.
Beaten Promotion Finalists Wood have been drawn at home to Dunfermline Athletic in round two.
Sutton, who Luke Garrard’s team beat to reach Wembley, will visit Airdieonians.
Sutton manager Paul Doswell said: “It’s going to test us in terms on the competitions we are in but the players prefer to play and not train! It’s something we’re delighted to be involved in.
“It’s a different type of team we’ll be playing as we are really looking forward to the matches and being part of something really unique.”
Wood boss Garrard said: “I think that it will give the players an added incentive and I am extremely excited that we have the luxury of possibly travelling around the British Isles for a fixture, I think that is a huge achievement.”