I think it's a great idea, since I have an undergraduate in EE and a masters in software. You will be more specialized, but also qualified for what I think are more interesting applications than people with a general software undergrad background. You will be less specialized than if you had a masters in EE.
Unless things have changed, EEs (electrical engineers) have a very solid background in math supporting electronics design, much of which is now done in software. I presently work with Boeing flight simulation software, where my control systems education ties into flight control of an airplane; it is done electronically 'fly by wire" with software instead of with hardware. I've also worked on many interesting types of medical equipment, where digital signal processing of ECG (electrocardiograms) was used, or image processing of microscopic cellular material, or ultrasound images, was used. DSP math fundamentals are taught in EE classes - Fourier analysis, LaPlace transforms, paving the way for z-transforms.
Companies like Google will be oriented more towards big data and networking. Networking does involve hardware, and I did just read an article about future Internet design with more routing done in software. Networks is certainly a growing area, but not really my specialty. They must have some hardware; they do drive around with those cameras on the cars.
Microsoft does produce some hardware; I knew someone who went to the hardware division to work on a mouse device. But there are many more companies than Microsoft and Google in the world. Microsoft has a wide range of software products.
I obtained my masters degree through an online program. It was originally NTU (National Technological University) but merged into Walden University. I don't know what the classes are like these days, but it should be available internationally. There are probably other online universities these days; it's a growing trend. Degrees from these might not be as prestigious, perhaps? Ask a prospective employer about this...
One of my employers, Philips, did have a section in India, although I have no idea which city. Philips is a Dutch company that owns many smaller companies, including several medical imaging companies.
I know I was also drawn to software; there is something very nice about changing your design and seeing results quickly and getting it into the customer product within weeks, instead of waiting months for new hardware. And the industry trend is to move functionality from hardware to software, so it seems like a good way to move. Trust your instincts and focus on what makes you unique and excel and sets you apart from your competition!