It was the second defeat in a row for the Ospreys following their narrow loss against the Blues on the opening weekend of the season and cost them their 15 month, 18 match unbeaten record at home.
Former Gloucester lock Eustace struck shortly after the restart as he scrambled his way over for a try which TMO Derek Bevan awarded after being consulted by the referee.
The first-half was a tense affair with only three penalties to get the scoreboard moving, two to home outside half Shaun Connor and one to his opposite number Ceirion Thomas. Then came that Eustace try that gave the visitors the lead, only for Connor to edge the Ospreys back in front with a third penalty.
Thomas replied after 64 minutes, after Morgan Stoddart had been high tackled by Lee Byrne, as the lead changed hands for the third time in 20 minutes. As the pressure mounted the Scarlets lost wing Darren Daniel to the sin-bin for a late charge on Byrne, but there was to be no denying the visitors.
Thomas landed a third penalty to stretch the lead and victory over their biggest rivals will have gone some way to make up for the home defeat at the hands of the Dragons.
