Wasps fielded a heavyweight side including Paul Sackey, James Haskell and took the lead when Mark Van Gisbergen kicked a penalty after just eight minutes.
But the Scarlets showed some fighting spirit and levelled up things ten minutes later when Wales man Stephen Jones.
However, home full-back van Gisbergen hit two more penalties and flanker Dafydd Jones was yellow carded. The Wasps lead was extended when Irish scrum-half Eoin Reddan crossed but - following sharp work by Simon Easterby and Darren Daniel - wing Mark Jones touched down to leave the Scarlets 14-8 down as the second half beckoned.
Jones and van Gisbergen struck more penalties before Wasps captain Raphael Ibanez was binned for a professional foul and Jones bagged another three points. Replacement 10 Thomas levelled the scores on 70 minutes and, after missing a late penalty, succeeded with his dramatic 30-metre winner.
The Scarlets begin their Magners League campaign against Ulster in Belfast next Friday