Josh Turnbull put the Ammies ahead after just two minutes, but a first-half brace by Stephen McLaughlin, including a penalty, put Mansfield in front.
In the 52nd minute, Matty Lund leveled as neither team could find a winner. The result eliminates Salford from play-off contention, meaning they will spend a fourth successive season in League Two.
Story of the match
The Ammies were in front within the opening 90 seconds of the match as Ibou Touray sent in a delectable ball that Turnbull headed past into the top left corner from inside the box.
In the eighth minute, Mansfield were nearly level, James Perch guiding a ball from the right-wing to the back post that found Matty Longstaff, who eluded his marker, but he went just wide.
The visitors were level on 15 minutes. Jamie Murphy drew a foul from Turnbull and curled his free-kick from 20 yards out around the wall and into the bottom left corner.
Ten minutes later, Rhys Oates stole the ball from Theo Vassell and drove at goal before Salford goalkeeper Tom King came out and denied the chance.
Mansfield shot-stopper Nathan Bishop came up with an enormous save after Brandon Thomas-Asante played in Matt Smith. The forward shot with his left foot only to see Bishop make the stop.
In stoppage time, McLaughlin won a penalty after his free-kick struck the arm of Turnbull inside the Stags' box. He stepped up to the spot and calmly slid past King to give Mansfield the lead.
Thomas-Asante would set up the hosts' equalizer seven minutes into the second half. His inswinging cross from the left was flicked on by Lund past Bishop.
Three minutes later, he nearly doubled his tally, nodding on Touray's corner from the center of the box, but it hit the crossbar.
King then saved from Oates, guiding the ball around the post with his left arm. He watched as George Maris curled one narrowly wide of the bottom corner minutes later.
At the other end, Thomas-Asante worked his way in from the left and his low shot produced a fine 74th-minute save by Bishop.
Neither side produced anything of note in the closing minutes as Salford saw their promotion hopes end while Mansfield have work to do to fulfill theirs.
Man of the match: Stephen McLaughlin
He did all he could to fire Mansfield into an automatic promotion place and his two goals on the day give him seven on the season, joint second on the club.