Wasps bound Josh Bassett crossed twice, his two tries decisive in helping the Blues to victory, but Llanelli's two first half tries were enough to worry the home side.
Playing with the slope in the first half, Llanelli were behind when Sharp gave the home side the lead with a 13th minute penalty. But they got back into it and two tries in four minutes put the visitors in the driving seat.
The first came from wing Kristian Phillips in the 23rd minute after Bedford coughed up possession. With men spare out wide, flanker Dan Thomas drew in the last defender before releasing Phillips to score under the posts. Dale Ford added the extra two.
Just before the half hour, Llanelli pulled further clear thanks to another well worked try. Scrum half Gareth Davies cut a superb line before playing a pinpoint pass to Nic Reynolds. The centre had enough pace to make the line to leave the home side shocked.
To their credit, Bedford refused to let their heads drop and they were soon back in the game thanks to Bassett. Bedford were piling into the Llanelli 22 and as the referee stuck his arm out for a penalty advantage, Sharp sent a chip over the defence which Bassett finished superbly in the left corner.
Sharp converted brilliantly before exchanging penalties with Ford as Llanelli led 13-15 at the break.
With a two point lead, Llanelli needed a fast start but Bedford went back into the lead through a Sharp penalty before Bassett got his second of the afternoon. Ford made a great cover tackle on Bedford centre Mark Atkinson but it was not enough to prevent him offloading to Bassett who crossed with ease.
Sharp converted but a Ford penalty in the 49th minute kept Llanelli within touching distance.
Llanelli tired in the closing stages though and when replacement Craig Cross yellow carded, three more Sharp penalties ended Llanelli's Cup run.