Do I need to apply for an ESTA application if my flight passes through U.S. airspace?

No. As long as the plane or ship you are on is not landing or stopping in the any continental United States territory, you do not require any ESTA authorisation (or visa, for that matter) as you won’t be passing through U.S. customs and immigration or setting foot on U.S. soil. This is generally true the world over when it comes to flying through a nation’s airspace. The only time you would need an ESTA is if you were landing on U.S. soil, even if it’s just for a stop over. You can have a look at this page for more information on stopovers.
However, if an aeroplane is passing, or potentially passing, through U.S. airspace, then the U.S. government requires travellers to provide information about themselves. This forms part of Secure Flight, an enhanced security check for passengers, and involves the United States being provided with your full name as it appears on your passport, your date of birth, your gender, and, if applicable, your redress number (a number provided to passengers by which, if they believe they have been improperly or unfairly delayed or prohibited from boarding an aircraft, to seek resolution and avoid future delays).
Secure Flight Passenger Data (SFPD) information is used to conduct watchlist matching of passenger information against the Terrorist Screening Database (TSDB) for all flights that overfly the continental United States, and is important for Homeland Security to prevent any people in the ‘No fly list’ from entering U.S. airspace. This implies that SFPD information will be required for travellers flying to or from Canada, Mexico, Cuba, and the Dominican Republic.
This information would need to be submitted at least 72 hours before your departure.
If you have provided your full Advance Passenger Information (API) while booking a flight, however, then the airline will provide the required personal details for you. This is just the information which you would have supplied when you bought your ticket. This system has been adopted by the US, most EU states, and a number of other countries. Airlines generally ask for this information during the booking stage or occasionally during check-in.
API consists of your full name as on your passport, gender, nationality, date of birth, travel document number (e.g. passport number), travel document expiry date, and travel document country of issue.
Your travel agent or airline will notify you well in advance that this will be occurring, and as long as you are not on any of the United States Transportation Security Administration (TSA) no-fly or other watch lists, this should not affect you.