How did the felines get to America?
Short answer
It is believed that low water levels occurring several times during the last ten million years facilitated migration and divergent evolution of the felids. The North-American ocelot, lynx, puma, leopard cat, and domestic cat are believed to have evolved from a common ancestor that crossed the Bering land bridge between 5 to 9 million years ago from Eurasia. Note that Pangea broke apart 175 million years ago, way before the Age of Mammals even started, 65 million years ago. 

Background
It is believed that modern felids arose in Asia with the divergence of the genus Panthera 10.8 million years ago (Mya) and the bay cat lineage 9.4 Mya. These dates correspond to the low sea levels that occurred during the late Miocene (Fig. 1). Between 8.5 to 5.6 Mya a progenitor of the caracal lineage arrived in Africa. The common ancestor to five felid lineages (ocelot, lynx, puma, leopard cat, and domestic cat) migrated across the Bering land bridge to North America, referred to as the New World migration. This period is also believed to have facilitated migrations of the Eurasian carnivores (ursid, procyonid, mustelid, and saber-toothed felids) to North America.

Additional Pliocene/Pleistocene migrations paved the way for the differentiation of the cheetah, originating from the North American puma and migrating to central Asia and Africa. Reversely, Asian Panthera species spread into America (jaguar, lion) and subsequently into Africa (lion and leopard). 


Fig. 1. Global migrations of felids. source: Johnson et al., (2006)  

Reference
- Johnson et al., Science (2006); 311: 73-7