AG2R's Pozzovivo made a solo dart for home from a small breakaway group around 2km from the finish and held off his charging companions to win by 3sec.
Colombia's Rigoberto Uran took second with Dan Martin of Ireland in third after 156.6km around Girona.
Pre-race favourite Alberto Contador finished safely in that chase group but fellow former Tour de France winner Chris Froome gave up around 20sec after failing to follow the initial break.
Thursday's fourth stage is the first mountainous one and leaves from Tona before arriving at the ski resort of La Molina 188.4km down the road.