The spaceship took off at 8:56 a.m. (1256 GMT) from Kennedy Space Center in Florida on a mission that will last 16 days and will include four days of work by its astronauts outside the ISS, to which it is also delivering needed supplies.
The Endeavor, with a history in space spanning 19 years and the newest shuttle in a fleet that had five space shuttles when fully operational, shed its two auxiliary lift-off rockets and continued into orbit without a hitch.
The congresswoman was gravely wounded in a January shooting at a Tucson shopping center, an attack that left six dead and 13 wounded.
After a stay at a Tucson hospital, Giffords was transferred to Houston's TIRR Memorial Hermann rehabilitation hospital so Kelly could be near her as he prepared for the shuttle mission.
Rounding out the crew were the mission's pilot, Greg Johnson, mission specialists Mike Fincke, Andrew Feustel and Greg Chamitoff, and the European Space Agency astronaut Roberto Vittori.
The astronauts of the Endeavor and the ISS will carry out a number of scientific experiments, including one using the Alpha Magnetic Spectrometer (AMS) on board the space shuttle that will make an astrophysics study of cosmic ray particles.