It was a dogged performance from the Scarlets and it was Stephen Jones who won the kicking battle over rival Andy Goode.
British Lions fly-half Jones booted 11 points including three penalties before converting Rhys Thomas’ 78th minute try to deny opposite number Goode victory in Pool Six.
The Scarlets were in desperate need of a win to rescue their reputation on the continent, having tasted success in just one of their previous 13 European ties.
Goode exchanged two penalties with Jones in the opening quarter before the England number ten added his third in the 27th minute before Lee Williams crossed for the opening try on the stroke of half-time.
Jones missed the conversion but showed his accuracy in the 53rd minute with a well-struck penalty to extend the home side’s lead to 14-9.
Replacement kickers Luciano Orquera and Rhys Priestland added further penalties before Wales prop Thomas crashed over to seal victory for the hosts.