Dario Gradi's men moved up one place in the table to 16th after a comfortable victory in difficult blustery conditions.
But even with the wind against him, Varney was able to despatch a terrific volley from wide right of the area into the far corner for his 19th goal of the campaign.
And his side could have added several more to that ninth-minute strike in a one-sided first 45. Maynard fired past the post and tested Cherries keeper Neil Moss with a stinging shot aimed at his near post.
Varney could have added a second in the 20th minute when Billy Jones and Maynard combined to put him in on goal, but the striker pulled his shot inches past the post as Moss came rushing off his line.
Maynard was unlucky not to make it 2-0 just over the half-hour mark when David Vaughan released him and he cut past a defender before curling in a great effort only to see it whack off the far post with Moss well beaten.
Paul Bignot fed Vaughan down the left and the Welshman's cross was just out of reach of being touched in. Then midfielder Gary Roberts produced a classy drag-back and thrashed a shot in at the near post which Moss did well to get behind.
Bournemouth offered little up front and their best chance fell to winger James Hayter, but he put a free header straight at Crewe keeper Ben Williams.
French defender Julien Baudet came close to opening his account for Crewe when he flicked Ben Rix's free kick inches past the far post on the restart. Then the centre-half whistled a free kick just over, while Rix drove a shot at Moss.
Moss did well keeping the Cherries in the game, but he was caught out when Crewe extended their lead in the 61st minute. When Young's clearance hit Hayter and rebounded to Bignot, the left-back drilled the ball across goal and, when the keeper could only parry, Maynard converted an easy chance from close range.
Bournemouth staged a late revival but it was too little, too late and they had no cutting edge. Substitute Brett Pitman lofted a difficult chance high over the bar after Darren Anderton flicked the ball into his path.
However it was left to Moss to keep the score down with a late double save when he blocked a shot from Roberts and then beat out Maynard's follow-up.