Hartlepool dominated the first half and deserved to be ahead in the 32nd minute. Richie Humphreys took a free-kick on the edge of the area which was deflected before going in off the inside of a post.
Swindon goalkeeper Rhys Evans made some important first half saves from Matt Robson, Lee Bullock and Michael Proctor, who spent some time last season on loan at Swindon.
Swindon had a great chance to equalise in first half injury time when Humphreys held back Jamie Cureton inside the area. But Cureton's tame penalty to the goalkeeper's right was easily saved at the foot of the post by Dimitrios Konstantopoulos.
Swindon were booed off at the interval but they were much improved after the break and the stronger side once Peacock had scored on 54 minutes. Aaron Brown centred from the left and Jack Smith got the ball back into the danger area for Peacock to head in from close range.
Evans tipped away another difficult free-kick from Humphreys and Proctor missed two good opportunities to have restored Hartlepool's lead.
But the best chance of a winner fell to the luckless Cureton with six minutes remaining. Brown got down the left and his low centre found Cureton six yards out only for the normally lethal striker to crash his shot against the underside of the bar when it looked easier to score.
Proctor, Thomas Butler and Darren Williams of Hartlepool were all booked as Hartlepool battled to survive.
Swindon had midfield player Charlie Comyn-Platt shown a yellow card for a foul in the 60th minute as Swindon battled to snatch a vital win.
Hartlepool had a great chance to win the match in the last minute after Proctor found himself with room to shoot inside the Swindon area but blazed wastefully over.