Oxford took the lead after just 9 minutes when Matt Paterson (pictured) went around young debutant keeper Ravan Constable to slot in an empty net.
Wealdstone replied 2 minutes into the second half when Evans Kouassi cut the ball back to David Pratt and he hit a well-struck low effort first time past Craig King.
Robbie Sinclair missed a golden opportunity to win the game for City in the 89th minute when they were awarded a penalty.
The former Forest Green Rovers man was tripped by Taofiq Olomowewe in the box, prompting the referee to point to the spot.
Sinclair stepped up himself to take the penalty but blasted his effort not only over the bar but also over the stand behind Constable’s goal.