Science, Military, and Economy could be thought of as the "triumvirate" in Civilization 5. Without a balance of all three, you're unlikely to be able to win. Most of the other systems in the game adjust this balance in some way. Not to say the others aren't important, and Culture, Happiness, and Growth could be considered next in line.
However, the point I'm trying to make is that without making Science a priority, you're unlikely to win. You'll find that many of the benefits of these other systems tend to feed back into Science, Military, and Economy.
Without Science, you can't build a strong enough fighting force to repel invaders. You can't build high-end economic buildings to increase your gold reserves and therefore have meaningful trade with other Civs. You can't unlock buildings to increase your growth, happiness, or production either. You also miss out on World Wonders, as by the time you can build them the other Civs have a major lead over you.
Science is central to the core mechanics of Civ5 (and all the Civ games, for that matter) and can't really be neglected without wrecking your entire Civ.
That said, you can also win by having more Military or Economic growth than your neighbors - although your question seems to indicate you'd rather not win via Military might alone.
You can win via Economy - a strong Economy and a Diplomatic Victory go hand-in-hand. Amass major amounts of gold and then buy out city states, and you'll quickly be able to win diplomatically. However, you've still got to keep your Science and Military balanced, lest you find yourself technologically outpaced or militarily overrun.
You don't mention Culture in your question, but Cultural victories are tailored towards smaller empires in Civ5 - if you wish to spend less time growing a massive Civ, you might consider a Cultural victory. With the Brave New World expansion, Cultural Victories are tied to Tourism, which is impacted by Diplomacy and Trade.