Will an "Excellent" throw give you a greater chance of capturing the Pokémon?

Getting a "Nice", "Great", or "Excellent" throw is possible by getting the Pokeball to land inside the shrinking colored circle. This gets you bonus experience if you catch it with that throw. The size of the circle determines which one you get. (Approximately, full circle-66%=Nice, 66%-33%=Great, 33%-smallest circle=Excellent.)

So getting those special throws won't boost your catch chance directly but having the circle as small as possible will.

The official FAQ says:

You have the greatest chance of capturing the Pokémon while the colored ring is at its smallest diameter.

After looking at many sources online and the research that others have done, making an "Nice", "Great", or "Excellent" throw does increase your chances of catching the Pokémon.

Source 1

Source 2

Looks like the throws bonuses are:

Nice Throw - 1.15X bonus

Great Throw - 1.5X bonus

Excellent Throw - 1.85X bonus

Here is a catch rate calculator so you can play with the odds yourself.