Calvin Zola had pointed to the way to a comfortable afternoon when he headed home early on, only for the Silkmen to hit back through giant Leicester loanee Ricky Sappleton.
Keith Alexander's side appeared to be heading for a deserved point when Grant slipped in to shoot into the bottom corner.
Gudjon Thordarson made three changes from the Alex side, which had gone out of the Johnstone's Paints Trophy in midweek. Dave Button replaced Adam Legzdins in goal after his return from Tottenham while there was a recall for Anthony Elding up front.
And Simon Walton was handed his first start in the middle of the park as Thordarson changed to a flat back four in the absence of the injured Mat Mitchel-King.
The Silkmen had former Crewe men Lee Bell and Colin Daniel in their starting eleven.
And they were soon on the attack when Sappleton laid the ball off for Hamza Bencherif to attempt a speculative chip from 25 yards which just cleared Button's bar.
But Crewe responded quickly and when John Brayford launched the ball over the box, Shaun Brisley needed to take evasive action with a timely header to put the ball behind out of Calvin Zola's reach.
New Macc loan keeper Jose Veiga was under orders to be tested it appeared as crosses from Jones and Brayford were thrown in.
Yet his team mates offered a genuine threat on the floor every time Izak Reid burst forward.
There was home anxiety when Matthew Tipton slid in Sappleton who crashed an angled drive into the hoardings.
But Zola was soon edging Crewe ahead in the 12th minute. When Daniel fouled Brayford out on the right flank, Jones arced in the resulting free kick and the Congolese striker's thumping header found the bottom corner despite Veiga getting a hand on the ball.
Bencherif failed to keep his first-time effort down following a neat lay-off by Daniel as Macc began to exert some pressure midway through the half.
Giant Sappleton was showing his strength when he bustled through down the right of the box, but Jones managed to nip in and divert the ball behind.
But Macc were nothing if game and Daniel drove inches over from 20 yards.
And they stunned Gresty Road with a 32nd-minute equaliser, which was hardly against the run of play. Sappleton latched onto Reid's service and when he spotted Button charging off his line he clipped the ball past the stranded keeper from way out on the right touchline.
Button redeemed himself minutes later when he clawed Ross Draper's header off the line.
But the visitors had their tails up and were seeing a lot of possession with Daniel blasting narrowly over.
Joel Grant had them retreating, though, when he moved into range and forced Veiga backwards to tip over a rising drive.
Then the keeper did well to hold Byron Moore's snapshot after Zola knocked down Walton's deep cross.
Sappleton turned on a Bencherif cross to test Veiga at the start of a second-half in which Crewe needed to show a major improvement.
But they were caught out down the right by old boy Daniel and when the winger pulled the ball back, Sappleton was inches away from hitting the top corner.
Zola should have put Crewe back in front in the 53rd minute when the goal opened up after Grant and Moore were blocked out from shooting. But the striker blasted wide from 12 yards out.
The lively Tipton responded at the other end when he drifted a shot past the far post.
Grant was breaking clear of Reid over the hour mark, but his finishing let him down as he wafted wide of the far post.
Then Veiga was behind Steve Schumacher's snapshot after a neat exchange of passes by Crewe around the visiting box.
Grant was emerging as The Alex's best hope as he drilled wide of the near post after making the shooting chance for himself.
When Moore was brought down, Jones was offered a shy at goal with one of his trademark free kicks. The Alex skipper was raising his arm to celebrate as his 70th-minute dead-ball effort beat Veiga and bounced down off the bar, but Macc were reprieved when the ball was adjudged not to have crossed the line.
Emile Sinclair sneaked in behind minutes later, but shot straight at Button before a better effort from the substitute tested Button low down at his left-hand post.
Thordarson threw on Aman Verma and Shaun Miller in the closing stages.
But it was Crewe's man of the match Grant who snatched the decisive third goal when he latched onto a long ball and beat Veiga at his near post with five minutes left.
Crewe: Button, Brayford, Jones, Ada, Worley, Moore, Walton (Verma, 75), Schumacher, Grant, Elding (Miller, 77), Zola.
Subs: Collis, O'Donnell, Murphy, Westwood, Davis.
Macclesfield: Veiga, Reid, Tremarco, Brisley, Morgan, Bencherif, Bell (Brown, 86), Draper, Sappleton, Tipton (Sinclair, 68), Daniel.
Subs: Brain, Rooney, Reed, Bolland, Dennis
Referee: Steve Bratt (West Mids)
Attendance: 4,151