Human evolution didn't start at the Big Bang: there was a very long period before the first stars were formed, and then two generations of stars were born and died before our Sun. After our Sun formed, it was even longer before the planets were created, and then for the Earth to cool down and become habitable. At that point, life itself had to form, and only then did the actual evolution happen.
Alien life probably couldn't have started before the third generation of stars, because of a lack of metals ("metals" in the astronomical sense meaning "anything that's not hydrogen"). But their planet could have become habitable much earlier, and the random processes that created life and then created intelligent life might have happened far faster through pure chance. All in all, it's been estimated that an alien civilization could be approximately a billion years ahead of us without any unknown chemical elements.
EDIT: To respond to your updated question, evolution doesn't have much to do with technology once you reach a certain level of intelligence. Genetically, modern humans aren't much different from the ancient Egyptians six thousand years ago. But in that time we've gone from the bare beginnings of agriculture to space travel. As soon as humans evolved a way to communicate abstract ideas, cultural evolution outpaced biology.