Your situation leave you less time to settle the issue because, you say the engagement is fixed.
Please ensure that you both are above the valid age for marriage and can prove it if need be, for the bride 18 years, for the bridegroom 21 years and both should be consenting without force( Hindu marriage act and Prohibition of Child Marriage Act).
Once you have done this, then think about the options and needs. Thinking of running away to get married is one thing but actually doing it successfully is a different ball game. If you have to get married on your own, then you need to look at the options of a registered marriage, have the documents for the same and witnesses to legalize your marriage.
Then you need to think of a place to live, a job or means to earn a livelihood, run a family and plans for additional expenses when there is a new arrival in the family. You need to have a plan in mind for all these and a back-up option. You need to face the wrath of the girl's family (verbal or physical abuse) or even a police complaint.
So, before you proceed, the girl and you should have a detailed and frank discussion. Be honest and brutally frank when it comes to having the commitment, plan for finances, accommodation and mandatory home expenses (often young lovers forget to grasp the financial needs and the stress of this adversely affects their love towards each other).
Once she is committed and you are willing to go ahead, then give on chance to both families, with a group of common friends and elders give have a discussion with them and try to find out the apprehension, is it cast,money, job security etc. Try to expalin and clarify their doubts. Then be firm and both of you can give a deadline and explain your choice to get married with their blessings, if not your willingness to be married otherwise.
You would be a better judge of the mood and the nature of the girl and her family, with this you can have a rough idea how the family are going to react to you. Accordingly have help form friends and dont' hesitate to seek legal help if you or both of you face violence from the girl's family.
If at all you go ahead with marriage without parent's consent, then have a mentor, friends and well-wishers from both sides to support you for the first few months or a year until the families forget the anger and animosity towards you and start accepting you both.