Cosgrove signed a new deal until the end of the season in the build-up to the game and it was his header in the 76th minute that proved decisive.
Crewe came within inches of taking a fifth-minute lead after a slick move involving Callum Ainley and George Cooper sent Jordan Bowery sprinting clear. He went round advancing goalkeeper Jack Bonham but slid his shot against a post.
Again it was the Alex forcing the pace as Eddie Nolan whipped a low ball across the face of the Carlisle goal with no-one on hand to convert the chance.
Carlisle skipper Danny Grainger powered a 25-yard effort inches past Ben Garratt's right-hand post and it was another defender, James Brown, who had the home side's best first-half effort when he got forward to hammer a shot from the edge of the area which flew narrowly wide.
Chances were scarce after the break, although Crewe almost profited from a 62nd-minute mix-up between Grainger and Bonham, but with the goalkeeper off his line, Cooper hoisted his lobbed effort onto the roof of the net.
Cosgrove had the ball in the net for Carlisle but was ruled offside. But the stalemate was broken in the 76th minute when Cosgrove headed Carlisle in front at the far post from a cross by Tom Parkes on the left.
A point-blank save by Garratt from Richie Bennett's shot prevented Carlisle from doubling their advantage in the closing minutes.