The Ibrox side took a first-half lead when Kyle Lafferty side-footed home from just inside the penalty area.
Hearts were always in the game when they were only one goal behind, but after substitute Saulius Mikoliunas was sent off for two yellow cards, then the visitors' chances virtually disappeared.
Rangers opened brightly and Boyd tested keeper Jamie MacDonald with a powerful shot from Andrius Velicka's flick.
Rangers new £3million signing Pedro Mendes made an impressive home debut, linking well and spreading play.
Rangers took the lead when Madjid Bougherra won the ball out on the right wing and from his cutback Lafferty dispatched a crisp shot low past MacDonald.
The second half started off with both sides going close. Mendes slid a superb pass through to release Boyd, but the Hearts keeper raced from his line to block the striker's shot.
Hearts were then unlucky not to equalise. Andrew Driver cut inside Kirk Broadfoot before unleashing a shot which Allan McGregor could only beat out, but the keeper redeemed himself with a marvellous save from Andrius Ksanavicius.
From a Mendes cross MacDonald palmed the ball away, but Boyd at the back post could only find the sidenetting.
Mikoliunas came on for the ineffectual Laryea Kingston, but within two minutes was booked for diving. Minutes later his substitute made a rash challenge on Kenny Miller and was red-carded for his second indiscretion.
Marius Zaliukas struck a powerful 30-yard shot which troubled McGregor before Rangers scored their second.
Miller cut past two players and advanced into the box before being brought down by Christophe Berra. Boyd drilled the resulting penalty low into the corner of the net.