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.
I have 3 reasons why the dwarves would win.
Firstly, Logistically the major advantage the elves had at weakening the dwarves' army was their archers as the dwarves advanced. The Dwarves took this advantage away with their siege weapons destroying the elves' arrows (what this siege weapon is I do not know) but it was extremely effective in mitigating the arrows.
Secondly, Take a step back, What were the elves fighting for? Their king's hurt pride and a couple of white gems? The elves were fighting only because their king told them too, And they knew it too. The dwarves, On the other hand, Were on their homeland, Fighting to reclaim the greatest dwarf kingdom that had ever existed; they were fighting for their fundamental right to exist.
Thirdly, We know that Dwarves were created from the stones of the Earth, And they possess super-strength. They are tough and doughy, Hard as nails, Wily as Scots, And take a great deal of punishment before they fall. Also, Fighting for Erebor, They seem to me to have luck on their side. If Earth gives them strength, Then this part of Earth loves them dearly; they will not fail. And they have battle goats.
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.