The visitors were ahead inside the opening 10 minutes as Rhys Oates was tripped in the penalty area by Andy Dawson, leaving the 21-year-old to hammer the ball into the bottom left corner from the spot.
© Getty Images
Salford responded well though, with the League Two side failing to clear a corner, leaving Chris Lynch with a free header inside the six-yard box, but his finish was a fraction wide.
The non-League club were level midway through the first half as Lewis Hardcastle launched the ball into the box, leaving Dawson to nod it into the path of Stephen O'Halloran, who smashed it into the back of the net from a couple of yards out.
© Getty Images
Salford saw plenty of the ball at the start of the second half, but it was Hartlepool who had the better of the chances, with Adam Jackson firing a loose ball wide, before Jay Lynch denied Oates a second.
The hosts were on the attack again on the hour mark as Richie Allen charged forward before squaring the ball to Hardcastle, but his low strike lacked power, leaving the Hartlepool keeper with a comfortable save.
The visitors wanted another penalty in stoppage time as Michael Duckworth went down under pressure from O'Halloran, but the referee was not interested, while the Salford goalscorer was furious with the Hartlepool defender's claims.
The result means that the teams will meet again at Victoria Park later this month, while both will be in the third-round draw, which takes place next week.