There were a wealth of chances at both ends over 120 thrilling minutes, but in the end Stags' top scorer Tyler Walker saw his penalty kick saved and Matt Dolan was able to seal the shoot-out 5-3 to send over 1,100 travelling Welsh fans into ecstasy behind that goal.
Nicky Ajose and CJ Hamilton netted their spot kicks before Walker's miss, then Mal Benning kept hopes alive as his kick went in off both posts before Dolan applied the final blow.
Newport were excellent in the first half and had hit the home bar twice before the Stags really came to life and Danny Rose had an effort cleared off the line when he looked a certain scorer.
The game became stretched and it was a major surprise there were no goals in the end with fans on the edge of their seats throughout.
But, ultimately, it wasn't to be for the Stags, who suffered a pre-match blow when veteran midfield ace Neal Bishop failed to recover from a calf injury picked up in the 1-1 first leg draw in Wales on Thursday and was replaced by Willem Tomlinson.
Newport also made one change with Joss Labadie in for Matt Dolan.
Stags started well and forced an early corner amid a superb atmosphere.
Then Hamilton hooked clear the first long throw of the game by Demetriou on three minutes as play swung from end to end in a thrilling opening.
Labadie was booked on eight minutes for his foul on Walker, but Mellis was well over with the free kick.
There was danger for Stags as O'Brien headed a free kick back across the face of goal with no one there to apply a finish.
On 12 minutes Logan had his first save to make as Sheehan tried his luck with a low shot from 25 yards after good hold-up play from Matt.
O'Brien was added to the book on 19 minutes for wrestling Walker to the floor after he had turned him.
The Exiles came close again on 22 minutes as Sheehan stretched for a forward ball from Butler and sent a volley inches wide into the side stanchion.
Two minutes later Matt found Amond in space to his right and he should have hit the target as he sent his low finish tamely into the sidenetting.
Turner got in a fine block to halt a blast from Willmott after Matt had set him up for a shot in the box as the visitors continued to look the more threatening.
The deadlock was almost broken from another long Demetriou throw on 39 minutes as the ball bounced in the box and Labadie headed against the bar from close range with Logan gratefully pouncing on the rebound.
Butler's challenge on Rose on 41 minutes saw Butler yellow-carded.
Newport missed another golden chance with 45 minutes up as Butler whipped in a powerful free kick and from six yards Amond sent his header over the bar with the goal at his mercy.
In stoppage time, in trying to head away a left wing cross, defender Turner saw the ball drop onto the top of his own crossbar as the visitors went in disappointed not be ahead.
Mansfield made an attacking change at the break as Hamilton was pushed up front with Rose and Walker with Benning coming on at left wing back and Tomlinson withdrawn from the midfield.
Immediately Walker made space for a shot that dinked just over the bar.
Then Hamilton pulled back a low ball from the left that Rose sent flying over the bar.
Bennett would have been relieved to see an offside flag up on 51 minutes after planting a six yard free header over the bar.
Mansfield struggled to clear another long throw, Logan eventually getting down to stop a low O'Brien shot.
Newport were desperately unlucky again on 59 minutes as Bennett let fly from 25 yard and saw his shot cannon back off the crossbar with Logan beaten.
But Logan was the hero two minutes later with a flying save to his left to keep out Sheehan's powerful 18 yard blast.
Mansfield's first on-target effort finally came on 64 minutes as Benning's free kick got through the wall but straight at Day.
On 67 minutes Turner twice came close, first seeing Day save his header from a Benning free kick, and when the ball was recycled back into the box, a second header also held by Day.
Day then had to push away an awkward shot from Walker on 71 minutes as the home side began to up the ante.
Labadie was hurt by a ferocious Turner tackle on 75 minutes and had to be replaced by Azeez.
An almighty scramble on 82 minutes ended with Logan clutching O'Brien's head-height drive as Stags survived again.
But Mansfield should have gone ahead on 85 minutes after Hamilton helped the ball towards Walker who lashed wide from 12 yards.
Three minutes later Rose flicked a header inches wide from a Benning cross.
Poole hacked down Hamilton in full flight on 89 minutes to earn a booking, Turner heading wide from the free kick, before Jones was also booked for a foul a minute into the added four as the 90 minutes ended goalless.
Atkinson replaced Mellis for the start of extra-time.
Stags were first to threaten as Hamilton got in down the left only to send his hurried cross-cum-shot well over.
On 96 minutes Hamilton might have settled the tie but, put into space by a great MacDonald pass, he blazed his finish over.
On 98 minutes Sheehan curled a powerful effort across the face of the home goal and wide of the far post from a tight angle.
There was high drama in the final minute of the first half of extra-time as Day saved superbly from a low Walker shot and Poole cleared Rose's follow-up off the line.
Stags' last throw of the dice was sending on Ajose for Rose as the game entered its final 15 minutes.
On the restart Stags could only clear a long Demetriou throw as far as Sheehan who volleyed wide from 18 yards.
Walker was again narrowly wide with a header from a corner and then went down under a challenge from Demetriou in the box with nothing given.
In the final minute Demetriou headed over a Willmott free kick before Stags sent on Grant for Jones as the game ended in penalties and heartbreak.
MANSFIELD TOWN: Logan; Pearce, Turner, Sweeney; Jones (Grant 120), MacDonald, Tomlinson (Benning HT), Mellis (Atkinson 91), Hamilton; Walker, Rose (Ajose 105). Subs not used: J. Smith, Khan, A. Smith.
NEWPORT COUNTY: Day, Poole, Demetriou, O’Brien, Willmott, Labadie (Azeez 75, McKirdy 110)), Sheehan (Dolan 110), Bennett, Butler, Amond, Matt. Subs not used: Pipe, Marsh-Brown, Randall, Crofts.
REFEREE: Michael Salisbury of Lancashire.
ATTENDANCE: 7,361 (1,118 away).
CHAD STAGS MAN OF THE MATCH: Alex MacDonald.