Goals from Arron Davies and Sammy Clingan either side of half-time steered Colin Calderwood's side to a 2-0 success.
But it could have been so much worse for Crewe, who saw Forest hit the post and bar on five separate occasions as Williams made a string of saves to keep the scoreline down.
Forest had been left needing to prove a point after being held to a 1-1 draw at lowly League Two side Lincoln in the FA Cup in their last match.
But they responded with their best performance of the season to extend their unbeaten run to five matches.
Forest were closest to drawing first blood as an inch-perfect ball from Chris Cohen split the Crewe defence and sent Nathan Tyson racing clear.
The striker let fly with a powerful, rising left-foot shot that Williams somehow got a hand to and directed it onto the bar and away to safety.
The home side were denied by the woodwork for a second time after Kris Commons had fed Lewis McGugan on the edge of the box.
The teenager left two defenders in his wake with some deft footwork but saw his crisp shot thud back off the inside of the post with a surprised Luke Chambers side-footing the rebound wide.
Calderwood's side were dominating but could not make the breakthrough with Cohen sending an audacious 45-yard lob narrowly wide after a Williams goal-kick had fallen short.
But the Crewe keeper was back on form in the 27th minute as he pushed a rasping volley from Davies onto the bar.
Crewe skipper Neil Cox had the ball in the back of the net in the 29th minute but the goal was rightly ruled out for offside.
And Forest finally earned the advantage their dominance had deserved on the stroke of half-time as Davies finally broke Williams' resolve with a rasping 20-yard shot that flew beyond the keeper and inside his right-hand upright.
That was Davies' first goal for Forest and six minutes after the break, Clingan netted his first for the club with a free-kick bent perfectly around the wall and into the bottom corner of the net.
Crewe were now rattled and, incredibly, Forest hit the woodwork for a fourth time as McGugan saw his shot palmed onto the bar by Williams.
On the hour mark, Forest hit the bar again as Cohen's shot was pushed onto the bar by Williams, who was having the game of his life as he made further saves to deny Davies and Clingan.