Midfielder Shakes broke the deadlock when he slotted home a left-footed strike from 16 yards just before the break for his second goal of the season. Striker Peacock doubled his side's advantage with a 25-yard curling free kick into the bottom right-hand corner.
Macclesfield goalkeeper Tommy Lee made a string of fine saves to keep his side in the game.
Lee's first save came after Christian Roberts beat off three challenges, but Lee ran out to bravely smother the ball at his feet.
Then the ex-Manchester United trainee brilliantly tipped over an Andy Nicholas header from point-blank range from Michael Timlin's corner.
Next, Lee made a fine reaction save after Swindon midfielder Sofiane Zaaboub had broken clear in the area. He dived at his feet and did enough to put Zaaboub off his shot as he fired it wide of the left-hand post.
Lee saved the best until last when he dived full length to tip Roberts' penalty around the left-hand post after the Robins front man won and then took the spot-kick.
Macclesfield threatened briefly in the first half when striker John Murphy latched on to a through ball but could only direct his shot straight at Phil Smith in the Swindon goal.
On 50 minutes Martin Bullock played in Alan Navarro whose shot from 18 yards was saved by Smith.
Substitute Matt Tipton, who came on in the 60th minute for Jamie Tolley, missed his side's best chance to get them back in the game. Tipton hit a right-footed drive that whistled past the post with Smith well beaten.
Swindon deserved all three points and if it was not for Lee's heroics in the Macclesfield goal they could have won by six or seven.