How can an Indian citizen travel to Ecuador on arrival visa from India without needing a transit visa along the way?

One option would be to transit in Amsterdam. KLM has routes to both Equador and India, and the Netherlands does not require airport transit visas of Indian nationals who stay within the non-Schengen transit zone of the airport.

None of the other destinations Wikipedia lists from Quito sound like they're likely to have direct flights to India and allow Indian citizens to transit visa-free. (Airport transit visas are required by Spain, France, Germany, and the UK; USA doesn't have visa-free transit for anyone). There may be more options with two layovers.

EDIT: Europe is more flexible than what I understood when I wrote this, as long as you are transiting airside only. However, I stand by my disclaimer that airlines are likely to cause problems (unjustifiably) if you have no visa for your transit country and also no visa for your final destination.

Well, it seems transit visas are required for the US and Europe unless you meet certain requirements. Getting a transit visa is surely the simplest option if possible.

If a transit visa is not possible at all, this is tricky because if you only stick to countries that don't require visas for Indian citizens, there aren't any flights. So we'll have to look for countries that allow flight transfers without a transit visa. Using WikiTravel's page on Avoiding a transit of the US, and Kayak's tool to find nonstop flight routes, it looks like there are a couple ways to do this:

EDIT (ADDED): India -> Amsterdam -> Ecuador (see Henning Makholm's answer)
EDIT (ADDED): India -> Frankfurt -> Panama City -> Ecuador (also some other variations on this one are possible, such as replacing Panama City by Mexico City or Bogota)

India -> Tokyo -> Mexico City -> Ecuador

• Tokyo doesn't require a visa for international transit as long as you are transiting on the same calendar day. (See: Layover at Narita)
• Mexico City doesn't require a visa for international transit if the connection time doesn't exceed 24 hours. (See: Wikipedia Visa policy of Mexico)

India -> Dubai -> São Paulo -> Bogota -> Ecuador

• Dubai doesn't require a visa for international transit of less than 24 hours. (See: Do I need a transit visa for UAE)
• Brazil doesn't require a visa for international transit if the connection is on the same ticket. Emirates Airlines sells tickets from Dubai to Bogota connecting in São Paulo. (See: Do Nigerian citizens need a transit visa for Brazil)
• Bogota doesn't require a visa for international transit of less than 24 hours. (See: Colombia.com: Visas y Permisos)

Disclaimer

Airlines are notorious for denying boarding incorrectly based on the agent's interpretation of what's in TIMATIC, and they will generally assume that you should have a visa for your final destination. If you're really planning to do this, it is crucial that you discuss it with the airline(s) in advance to make sure they will allow you to board.