Dragon Quest invented the wheel. DQ1 was the first and quintessential JRPG, heavily influencing the development of not only FF1 but every other early JRPG. The DQ series as a whole symbiosed with later entries of FF and the two franchises evolved together and played off each other mechanically and artistically. And just go read about the history of Dragon Quest and EarthBound.
I knew about Final Fantasy long before I ever heard of Dragon Quest. FFX was one of the first RPGs I ever seriously played, and to this day is my favorite JRPG. But once I learned about the influence of Dragon Quest, I resolved to play at least a few of them, and I exhort everyone interested in RPGs and their development to do like I did. It was a good experience.
You can't have FF without DQ, but we did, for a short time, have DQ without FF. So, even though I like Final Fantasy better on a personal level, I have to give it to Dragon Quest.