Yes, India does make rocket engines. Examples are Vikas engine, which is used in PSLV, CE-20 cryogenic rocket engine which is used in GSLV rockets of ISRO. You can read about the GSLV Mk III – A Comprehensive Overview of Heavy Lift Launch Vehicle in the given link.
- Indian Space Research Organisation [ISRO Notes for UPSC]
- Mars Orbiter Mission (MOM) – A Complete Overview of Mangalyaan Mission