Crow, who struck a late equaliser to rescue a point against Yeovil on Tuesday night, headed Northampton into a third-minute lead, lunging to meet a left-wing cross from Martin Smith.
That was enough to decide a niggly affair against a Macclesfield side whose eight-match unbeaten run came grinding to a halt.
The Silkmen's 23-goal top scorer Jonathan Parkin had one of his off days in front of goal.
In the first half, he was denied by a smart save from Lee Harper but he should have squared things up four minutes after the restart when he was sent racing clear on Harper's goal, but somehow he could only shoot into the side-netting from 12 yards.
Seven minutes later he used his body strength to shrug aside Fred Murray but Harper was equal to his low shot.
Buoyed by their early goal, Northampton could have added to their lead but Charlie Hearn saw his spectacular 29th-minute volley cannon off the inside of a post before Tommy Jaszczun hit a rising drive from 30 yards which just cleared the crossbar.
In the second half, Macclesfield must have sensed a sinking feeling when Parkin missed the target and then Danny Swailes saw his goal-bound header from Kevin McIntyre's corner cleared from under the crossbar by Smith.
At the other end, the Cobblers could have given themselves a cushion as they strengthened their play-off position but Smith saw his close-range shot saved in unorthodox fashion by the legs of goalkeeper Alan Fettis, the ball bouncing up and over the crossbar.
The victory sets Northampton up for their local derby at Rushden & Diamonds next Saturday.