Dinosaurs lived during the Mesozoic Era which consists of three periods: the Triassic, Jurassic and Cretaceous. In general, the world was a much hotter place when dinosaurs lived because of high levels of carbon dioxide in the atmosphere. This caused a mini-greenhouse effect which kept the planet lush and green. Temperatures were so high on planet Earth that the polar ice caps completely melted away. These were ideal conditions for our reptilian friends.
Because the Earth was so hot back then, dinosaurs had an extremely wide geographic range. Dinosaur remains have been discovered on every continent, and nearly every latitude. Cold simply was not a limiting factor for these animals.