Check out Air Arabia, www.airarabia.com .

They offer one-way tickets from Mumbai to Sharjah (which I believe is just outside of Dubai) for appr. 5,000 IR.

I can't offer any insight into the quality of their operations. Perhaps someone else on this forum has travelled with them and can provide info.

Air India Express, www.airindiaexpress.in/airindiaexpress/home.asp , also offers flights from several points in India (not Mumbai or Chennai) to Dubai.