EXHIBIT 32.1

The following certifications are being furnished solely to accompany the Report pursuant to 18 U.S.C. § 1350 and in accordance with SEC Release No. 33-8238. These certifications shall not be deemed filed for purposes of Section 18 of the Securities Exchange Act of 1934, as amended, nor shall they be incorporated by reference in any
filing we make under the Securities Act of 1933, as amended, whether made before or after the date hereof, regardless of any general incorporation language in such filing.

Certification of Chief Executive Officer

Pursuant to 18 U.S.C. § 1350, as created by Section 906 of the Sarbanes-Oxley Act of 2002, the undersigned officer of ResMed Inc., a Delaware
corporation (the Company), hereby certifies, to his knowledge, that in connection with the Annual Report of the Company on Form 10-K/A for the year ended June 30, 2016 (the
Report):



|(i) | the Report fully complies with the requirements of Section 13(a) or Section 15(d), as applicable, of the Securities Exchange Act of 1934, as amended; and
|--- | ---------------------------------------------------------------------------------------------------------------------------------------------------------



|(ii) | the information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Company.
|---- | ----------------------------------------------------------------------------------------------------------------------------------------------------



|                             
|-----------------------------
|Dated: January 12, 2017      
|                             
|/s/ MICHAEL J. FARRELL       
|Michael J. Farrell           
|Chief executive officer      
|(Principal Executive Officer)

A signed original of this written statement required by Section 906 has been provided to ResMed Inc. and will be retained
by ResMed Inc. and furnished to the Securities and Exchange Commission or its staff upon request.

Certification of Chief Financial
Officer

Pursuant to 18 U.S.C. § 1350, as created by Section 906 of the Sarbanes-Oxley Act of 2002, the undersigned officer of ResMed Inc.,
a Delaware, corporation (the Company), hereby certifies, to his knowledge, that in connection with the Annual Report of the Company on Form 10-K/A for the year ended June 30, 2016 (the
Report):



|(iii) | the Report fully complies with the requirements of Section 13(a) or Section 15(d), as applicable, of the Securities Exchange Act of 1934, as amended; and
|----- | ---------------------------------------------------------------------------------------------------------------------------------------------------------



|(iv) | the information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Company.
|---- | ----------------------------------------------------------------------------------------------------------------------------------------------------



|                                                              
|--------------------------------------------------------------
|Dated: January 12, 2017                                       
|                                                              
|/s/ BRETT A. SANDERCOCK                                       
|Brett A. Sandercock                                           
|Chief financial officer                                       
|(Principal Financial Officer and Principal Accounting Officer)

A signed original of this written statement required by Section 906 has been provided to ResMed Inc. and will be retained
by ResMed Inc. and furnished to the Securities and Exchange Commission or its staff upon request.