Ulster led 17-3 at halftime through tries by full back Jamie Smith and wing Tom Seymour, both converted by outside-half Paddy Jackson who also kicked a penalty as against one for Neath by outside-half Dai Langdon.
But the second-half proved a different story as the All Blacks staged the comeback of all comebacks to register three tries from skipper Gareth Gravell, prop Ross Davies and winger Kevin James. Langdon converted two and kicked a penalty, Jackson kicking a goal for Ulster.
Neath now line up against group favourites Bedford at The Gnoll on Wednesday (KO 7.15pm).
