The Scottish side had led 14-0 at one point and 17-13 at half time but the Blacks battled back with a three try second half salvo.
Stirling started well with early tries from Alex Moffett and Adam Welsh, both from good forward platforms. Bryan Archibald converted both scores.
Neath hit back with a try from Kevin James, the imposing winger half charged down a Stirling clearance and plunged over. Dean Cummins added the extras.
Stirling saw yellow when Benji Passanti was penalised for persistent infringement and Cummins closed the gap to four points. Rhys Jones was then given 10 minutes for a high tackle as Archibald and Cummins exchanged penalties before half time.
Despite Owain Ford suffering a serious looking injury just before the break, the second half started well, Luc Jones breaking clear to score near the posts as Neath led for the first time.
A raft of changes in the next fifteen minutes saw the game’s momentum interrupted before Cummins added another penalty to his tally and Chris Morgans broke clear to score a tremendous team try.
The bonus point score came following a flowing move on 75 minutes, Lloyd Phillips landing on the ball after confusion near the visitors’ line. Cummins missed the kick, his only failure in a 15 point contribution.
Stirling got a late consolation through back rower Steve Swindall, but it was undoubtedly Neath’s day.
In the other two games to take place involving Welsh clubs, there were also wins for Swansea and Carmarthen Quins.
Swansea faced a difficult trip north to take on Dundee but were rewarded for their adventure with a narrow 5-11 win.
Scott Otten crossed for the decisive score in the first half and a goalless second period saw Mefin Davies’ side hold on for the win.
In the other Pool 3 fixture, Carmarthen Quins saw off Cornish Pirates with an impressive 17-12 win.
Wales Sevens man Jason Harries was once again on the score sheet, fly half James Dixon kicked three penalties and Jack Maynard added one of his own as Quins toppled the pool winners.
