I understand that successful can be a subjective term, however, you can interpret it in your own way i suppose. Also, when I say Spanish Empire I'm speaking to dates ranging from the 15th - 19th centuries. All answers are greatly appreciated! Thank you in advance.
Arguably Hernan Cortes' Caravel -- true it didn't win great battles, and Cortes seems to have burned it on the beach, but this is the vessel that delivered Mexico to the Spanish. Similarly, Pizarro's ship, whose name I can't find in my sources. Christopher Columbus might merit a mention, but insofar as claiming land for Spain that stayed Spanish, Cortes and Pizarro did better . . . one might also count Magellan's flagship, the Trinidad. Magellan didn't complete the conquest of the Philippines, but in terms of setting the stage for Spain as a global maritime empire, he does well.
In terms of naval battles, Spain's greatest naval victory was at Lepanto, 1571. Don John's flagship there was a galley called the La Real (at the time, all Spanish flagships bore that name, it just means "The Royal"), which would be a contender. A replica exists in Barcelona's maritime museum.
Spain also would have any number of ships that served the empire by successfully hauling gold from the Western hemisphere back to Seville. We don't think of gold transports with same grandeur as, say, Nelson's Victory - but that's what made the Empire run. So you might say that "the most successful Spanish warship" was an entire fleet, the flota de Indias, sometimes known in English as "the plate fleets" which despite the enthusiastic attempts by privateers and pirates, generally delivered the huge mass of treasure back to Spain safely.