The Irish province proved too good in every department to run out 35-19 winners.
Neath were left to rue basic mistakes that handed the visitors two simple tries in the first half, which proved the difference in the end.
All Blacks coach Lyn Jones now has the task to get his side up for next weekend’s difficult away trip to Edinburgh, where five points could sneak his side into a home quarter-final.
Fly-half Lee Jarvis kept Neath in touch with four penalty goals, but his side’s reputation as the best defensive team in the competition took a battering as Munster ran in four tries.
In Galway, Connacht’s unbeaten run in the Celtic League came to a crashing halt at the hands of Pontypridd.
The men from Pontypridd were in superb form, running out 40-0 winners against a side that had never been beaten by a Welsh team in the Celtic League.
Winger Gareth Wyatt scored a brace of tries and veteran fly-half Neil Jenkins chimed in with 12 points with the boot.
A late pair of tries to replacement back Matt Allen has put Cardiff into the Celtic League quarter-finals.
The Welsh capital beat Bridgend 37-12 at the Arms Park, and their bonus point for four tries puts them into the playoffs.
It was Cardiff’s fourth win on end after a poor start to the season, while Bridgend crashed to their fourth consecutive loss after starting with wins over Newport and Borders.
Nicky Robinson (Cardiff) and Craig Warlow (Bridgend) each scored 12 points with the boot, but Cardiff’s star-studded backline proved the difference on the day.
In contrast, the Ravens were clueless in attack and never looked like crossing the home try-line.
Edinburgh confirmed their place in the final eight with a 30-20 win over a gritty Ebbw Vale side, with talented scrum-half Mike Blair scoring a late try to give his team some breathing space.
The visitors to Eugene Cross trailed 10-13 at the break before British Lion backrower Simon Taylor and Blair scored crucial five-pointers.
And just when long-suffering Caerphilly fans thought their season couldn’t get any worse, it did. Irish province Ulster left Virginia Park with a crushing 67-15 victory, with Ireland fly-half David Humphreys kicking 27 points.
