Lee Fowler's close-range finish and a penalty from Connor Jennings gave Wrexham a commanding 2-0 half-time advantage.
Kayden Jackson hit the post as Wrexham continued to press after the break but were unable to add to their lead.
Wrexham are now sixth in the table and only behind Gateshead, in the final play-off spot, by two points.
Manager Gary Mills told BBC Radio Wales: "We've played well today and in the second half I thought we were outsanding.
"It was an excellent performance. It's never easy coming here, we know that, and I think it's the first time I've won here actually and it feels good."
Wrexham beat Welling to move within touching distance of the National League play-offs.