The young Magpies led at half-time, courtesy of Neave's penalty at United's Little Benton Academy, before the hosts conceded twice in the space of five second-half minutes following quick-fire strikes from Joe Bevan and Jesse Williams.
Joe Brayson, introduced from the bench, levelled the scoring with a composed finish before Neave secured maximum points in the dying stages, netting his ninth goal of the season at under-21 level in the process, after turning the ball home from Leo Shahar's inviting cross.
Adam Lawrence made five changes from last week's narrow 1-0 defeat against table toppers Ipswich Town as Ben Parkinson, who made his Premier League debut for Eddie Howe's side away at AFC Bournemouth in November 2023, was handed his first competitive start at second-string level since April following a long-term foot injury.
The hosts almost got off to a strong start as Anthony Munda raced onto Harrison Ashby's threaded pass in behind the Clarets defence but the former's low cross into the six-yard area was hacked out for a corner.
Neave hit the woodwork shortly after as the Winlaton-born forward, named on the bench for Newcastle's 2025 Carabao Cup final victory against Liverpool, collected Alfie Harrison's pass and after his initial shot at goal was blocked, the 18-year-old's resulting drive at goal brushed the crossbar.
Newcastle's youngsters took the lead in the 35th minute following Neave's spot-kick. After latching onto Harrison's dangerous pass inside the box, the England Under-18 international was bundled over inside the penalty area by Joe Ashton before converting the resulting penalty into the bottom left-hand corner, sending Burnley goalkeeper Oisin Cooney the wrong way from 12 yards.
Magpies goalkeeper Max Thompson produced an impressive stop to deny Vernon Masara after the restart as the Burnley forward initially raced in behind the Magpies defence but the United shot-stopper stood tall to turn the attempt at goal out for a corner with a strong left arm.
In the 58th minute, though, Burnley drew level through Bevan as the Clarets striker picked up the ball on the right flank and, after dangerously cutting inside, his deflected effort flew into the far left-hand corner.
Newcastle suffered a further setback in the 63rd minute through Williams as the Burnley skipper latched onto Hannes Delcroix's knockdown from an in-swinging corner before rifling the ball into the roof of the net from close-range.
The hosts found themselves level in the 78th minute through Brayson as the 18-year-old striker, introduced at the restart in place of the returning Parkinson, controlled Munda's pass into the box before calmly slotting a low effort beyond Cooney.
With the full-time whistle nearing, Neave sealed Newcastle's first Premier League 2 win of the season on home soil in the second minute of stoppage-time, connecting with Shahar's excellent pass from the right wing before guiding the ball into the bottom left-hand corner, staging a dramatic fightback in United's final league game of the calendar year.
Newcastle United Under-21s: Max Thompson, Harrison Ashby (Leo Shahar 46), Miodrag Pivaš, Cathal Heffernan (c), Alex Murphy, Scott Bailey, Anthony Munda, Sean Neave (Charlie McArthur 90), Alfie Harrison, Dylan Charlton (Seung-soo Park 63), Ben Parkinson (Joe Brayson 46).
Subs not used: James Taylor.
