Nordi Mukiele scored the decisive goal in the 34th minute after a superb combination with Trai Hume and Brian Brobbey, coolly finishing from a tight angle.
Manager Regis Le Bris welcomed Robin Roefs, Habib Diarra and Noah Sadiki back into the starting line-up following extended breaks after their World Cup involvement.
The Black Cats dominated possession and created several chances, with Diarra and Romaine Mundle threatening early on.
Wrexham briefly thought they had equalised through Nathan Broadhead, but the goal was ruled out for offside.
Nilson Angulo struck the crossbar after the restart, while Trai Hume also went close. Mukiele’s first-half effort ultimately proved enough.
The Black Cats next face RC Lens on Saturday, August 8th.