And had the Dragons been more clinical in front of goal, they would have celebrated an emphatic victory.
It was the Conference side who started brightest and they should have capped their early pressure with a goal, when leading scorer Steve Morison headed weakly at Michael Ingham.
The hosts slowly took control of the game and Williams twice went close before he broke the deadlock four minutes from half time.
Darren Ferguson's corner was not properly cleared and when Steve Evans played the ball back in to the danger area Williams was on hand to direct his header into the top corner.
Wrexham dominated after the restart with Evans and Kevin Smith going close but Stevenage could have earned a replay after a stirring late fightback during which Morison headed wide and substitute Craig Dobson had two good chances.