There's always a way! My friend, you are not just normal; you're special! You're special because you wish to pursue a masters degree in engineering. You're special because you don't want to settle for a mediocre job. You're special because you are concerned about your career, don't want to leave it to chance, and are ready to work hard to shape it the way you want. And you're certainly special because you took the pain, and made the effort of posting your question on Quora. Not many people are the same. So let's settle on this: you're special! :)
Coming to facts, while 60% is not a bad score, percentage is not the only criteria to get admitted to a Masters program at IITs, NITs, and other colleges (you missed mentioning IISc!). For admission to these colleges, your GATE score would supersede the marks obtained in your bachelors. Only if you apply for a MSc(Engg)/MS or a PhD, your marks come into picture (that too, very lowly weighted). Now I'll split the answer based on your (expected) graduation year (but please read all points: 1, 2, 3):
1. Before 2014:
If you've graduated before 2014 (i.e., in the year 2013 or before), you might either be employed in a job presently, or might be looking for one.
Now, if you have taken the 2014 GATE exam held this year in February, you would know your GATE rank. Expect to get admitted to ME/MTech in IITs/IISc only if you have a rank in the range 1-300 (the cutoffs vary from department to department, and they do depend on your Category too). For NITs, the cutoffs are even higher (it's a good exercise to look for previous years' GATE cutoffs for different colleges). Although the admission deadlines are over, application to a MSc(Engg)/MS or a PhD program is also highly advisable.
If you haven't appeared for the GATE exam this year (2014), or if your result is not adequate for you to get an admission, prepare sincerely for GATE 2015 (join a coaching institute, if possible). Forget marks!
Note: I don't advise you to leave your job straightaway and go for a masters, but if you can afford to do so (if you'd be financially stable for the next 2 years), just go for it. You'll have difficulties doing it later.
2. In 2014:
If you're going to graduate this year (in 2014), you might be at the end of your fourth year. You would either have already been placed, or would be looking for a placement. But since you've decided to go for a masters, don't worry about placements.
Now comes the question again: Did you appear for GATE 2014? If yes, read point#1, since you'd have already got your results.
If not (or if you aren't satisfied by your GATE result), then prepare for GATE 2015. You have one year for the preparation. If possible, join a coaching institute and give the preparation your best bet. Meanwhile, you can join as an employee in a company, or can join as a teacher in an academic institution, or apply for a one-year internship/fellowship/research-assistantship in a research lab at IITs/IISc/NITs/IIITs for an interesting project. This will keep you occupied. Even if you don't find a job/internship, don't worry. Prepare for the GATE exam, and keep educating yourself about higher education: the prospective areas you can work on. Also, try to learn new technologies in the one year (e.g., a new programming language, or a new MOOC on Coursera). Just keep the spirit alive!
3. After 2014:
This means, you are a third year (or a second/first year) student. You have more-than-enough time to prepare for the GATE exam. If you want to be strategic about the preparation, join a coaching institute; however, self-study may suffice, since the topics would be fresh in your mind. Don't worry too much about the marks you have got (or, will get). Try to do your best in the rest of the exams, and take the GATE exam very seriously. It's the ticket to enter IITs/IISc/NITs (the only one for an ME/MTech).
EDIT: Based on the comment, I'd like to add that GATE will also open opportunities for a masters at NUS/NTU (Singapore). Though I'm a little skeptical about going for a masters outside India (see my answer to Higher Education: Which is better? IISc, Bangalore or NUS, Singapore.), especially if you get a call from IISc or some of the IITs or one of TIFR/CMI/ISI for the same, but your decision should be your own. Make best use of the choices you have. :)