New American Success Act of 2014 - Establishes in the Executive Office of the President a National Office of New Americans which shall:

• coordinate federal, state, and local efforts to ensure the economic, linguistic, and civic integration of immigrants and their children;
• advise the President, Congress, and other federal officials on immigrant integration;
• establish national goals for immigrant integration;
• cooperate with the Office of Management and Budget (OMB) and other executive agencies to analyze the budget impact of immigration policies and integration efforts;
• identify the anticipated effects of new federal immigration policies on existing integration efforts; and
• consult on a biannual basis with state and local government officials.

Establishes within the Office a Task Force on New Americans.

Requires U.S. Citizenship and Immigration Services, in consultation with the Office, to award initial entry, adjustment, and citizenship assistance grants to local government units, private organizations, community-based organizations, or not-for-profit organizations to provide legal assistance to an eligible non-citizen who is seeking: (1) to become a permanent resident or naturalized citizen, or (2) relief from removal.

Makes such grants available for 10 fiscal years beginning with FY2016.

Establishes in the Treasury the Integration Success Fund.

Amends the Immigration and Nationality Act to revise the waiver of the English language naturalization requirement for senior individuals.