The 20-year-old forward, who has been with the Stags academy since 2018, netted his first professional goal in the 84th minute of the game, cancelling out Joss Labadie's earlier opener.
The result prevents the away side from climbing above Forest Green Rovers and Morecambe in the Sky Bet League Two table, as the automatic promotion chase heats up with around seven games remaining of the season.
Mansfield dominated the contest, having 55% of possession and 18 shots compared to Newport's seven.
Mansfield Town: Stone; Perch, Rawson, Sweeney, McLoughlin, Clarke, Maris, Lapslie, Quinn, Sinclair, Bowery.
Newport County: Townsend; Bennett, Dolan, Demetriou, Shephard, Labadie, Sheehan, Ledley, Lewis, Taylor, Telford.
Story of the match
Newport got the game underway as the more dominant side, as full-back and former Swansea City prodigy Aaron Lewis took aim from distance, but Aidan Stone was well versed to the challenge.
Mansfield responded strongly however, as a deep cross found Sinclair's head, which he arrowed against the post and was denied an early opener.
The Exiles almost shot themselves in front courtesy of Dom Telford, as 24-year-old had his near post strike thwarted by Stone.
Heading into the second half, Newport continued their dominant spell as Sheehan found a yard of space and fired across goal, but Stone was prepared and diverted his cross-shot away from danger.
With both sides desperately hunting an opener, it was Mansfield that came unbearably close, as Stephen Quinn finished off a brilliant move from the Stags by hitting the inside of the post with his shot. The ball teasingly rolled back into the danger area, but was smashed clear to spare Newport's blushes.
However, the game wouldn't be finishing goalless, as Newport captain Labadie pounced onto some poor control from George Maris, before he raced into space and blasted a low strike beneath Stone, who will be disappointed he didn't keep it out, before reeling away in jubilance. A third of the season for the 30-year old.
But there's a reason Mansfield have lost just one of their previous seven games, as it was time for academy graduate Sinclair to shine. The 20-year-old bravely stole possession from Sheehan deep in the box, before he drove towards the six-yards box and rifled a strike into the roof of the net, leaving Nick Townsend little chance in keeping his clean sheet. A proud moment that Sinclair will remember for the rest of his life.
With the late drama unfolding, the referee called the game to a close with the points being shared in Nottinghamshire.
Play-off worthy Newport?
Manager Michael Flynn will be disappointed with his side's underwhelming performance and inability to defeat a Mansfield side whose last victory came in early March. Newport are a side who will be worryingly looking over their shoulders at the chasing pack, so if they are to keep in the play-offs and be involved in the road to Wembley, the Exiles need to get their heads together and get back to the riveting football they can play.
Glimmers of hope for Mansfield
Whilst this season has been less than celebratory for Mansfield, manager Nigel Clough should be pleased with his side picking up a point against a side chasing promotion. With youngsters like Sinclair shining in the first teams squad, it shows reasons for hope and anticipation for next season, when Clough can really build his own squad and get this Mansfield side playing the way he wants.
The 35-year-old attacking midfielder pulled the strings for Mansfield in midfield, being a nuisance when opposing the Newport midfield and defence and creating numerous chances. He even came close to scoring himself, when striking the post.