First off, dwarfs have much stronger armor and a better variety of weapon than the elves do. Second, if the elves had shot a couple volleys of arrows than the dwarfs probably could have blocked them well with there shield formations and superior armor. Lastly, the dwarfs would have crushed elven lines with an armored charge like they did against the orcs in the third hobbit. Thank you.
They would win because they have armor that it denser, and therefore stronger then the Elves, because the Elves value speed and cunning, long distance attacks, and/or magic to fight. Dwarves also have less height to cover with armor, meaning the more metals could be used for actual protection instead of coverage.
Final reason. The Underdogs always win.
It has been said that the dwarves have better armor. What evidence is there for this claim? As the First Children of Iluvatar, Elves have had millennia to perfect their armor. Although light, it is strong. In addition, they have had centuries to hone their combat skills.
Elves are just as efficient in long-distance battle as hand-to-hand combat. Just look at Thranduil, who didn't even carry a bow in the movies. He did very well with just his sword alone. Lastly, Elves are agile and fast (not to mention the height advantage they have over the dwarves).
In summary, elves have the skill and armor necessary to destroy whoever stands in their way, whether it be dwarves, orcs or Uruk-Hai.