Gareth Evans set the ball rolling for the hosts at Highbury after 22 seconds, before goals from Chris Clements and Anthony Howell put Mansfield in front.
Fleetwood then pulled level through Antoni Sarcevic and were two goals ahead by the 66th minute thanks to a Sarcevic penalty and another Evans strike.
Samuel Clucas and Ross Dyer struck back for the visitors to leave the tie all square after 89 minutes, before Sarcevic sealed his hat-trick and the victory with a second spot-kick in the sixth minute of stoppage time.
The narrow victory leaves Fleetwood in fourth, one point behind leaders Oxford United, who strolled to a 3-0 win at home to Morecambe.
First-half goals from James Constable, Sean Rigg and Ryan Williams were enough for Chris Wilder's men, who remain ahead of Chesterfield and Rochdale on goal difference.
Oliver Banks and Armand Gnanduillet scored as Chesterfield recorded their first back-to-back victories since the end of September in a 2-0 win over Wycombe Wanderers, while Michael Rose, Graham Cummins and Andrew Tutte found the net in Rochdale's 3-1 win against Exeter City.
Elsewhere in the chase for the top spots, Southend United overcame York City 2-1 as both sides ended the match with 10 men, Newport County were held to a goalless draw by Cheltenham Town and Burton Albion beat Bristol Rovers 1-0 in a clash that saw three players sent off.
At the other end of the table, defeats for Northampton Town and Torquay United left them languishing in the relegation places.
Aidy Boothroyd's Northampton lost 2-0 at Hartlepool United, while Torquay allowed Accrington Stanley to drag themselves clear of the drop zone in a match that ended 2-1.
In the day's remaining fixtures, Bury picked up a valuable point in a 1-1 draw with AFC Wimbledon, Scunthorpe United kept up the pressure on the play-off positions with a 2-1 win at Portsmouth, and Plymouth Argyle edged to a 2-1 victory over Dagenham and Redbridge.
