The United States international netted the only goal of the game early on as the Ibrox side secured their fourth win in five meetings with Billy Reid's men this season.
The victory was hardly a convincing one but allowed the Scottish champions to take another step closer to the championship ahead of Celtic's clash with Hibernian on Sunday.
Today was Heroes and Legends Day, where Rangers paid tribute to former stars from the 1950s to the 1990s with the likes of Willie Henderson, Ralph Brand, Sandy Jardine and Andy Goram parading the League Cup at half-time.
And the class of 2010 were well aware that if they wanted to remain on course to add the SPL title to their trophy haul this season they had to get back on track with a win against Hamilton.
Walter Smith made one change from the side who suffered a shock 4-1 defeat at St Johnstone midweek, with Edu replacing Kevin Thomson in the midfield.
Accies were boosted by the news that top goalscorer Mickael Antoine-Curier was fit enough to start despite being rated doubtful with an ankle injury, while the suspended James Wesolowsi was replaced by David Elebert.
Rangers knew they needed a positive reaction after the events in Perth on Tuesday night and could have taken the lead in the opening minutes.
Kenny Miller shrugged off both Martin Canning and Mark McLaughlin to unleash a shot from a tight angle that was blocked at the near post by goalkeeper Tomas Cerny.
Edu was then denied by the woodwork when his powerful low drive from all of 35 yards crashed off Cerny's left-hand post.
But the American was celebrating seconds later when he ran onto Miller's pass before slotting past the Accies keeper from just outside the six-yard box with 11 minutes gone.
Despite an impressive opening spell, Rangers failed to build on their lead and Hamilton gradually began to impose themselves on the game.
They could have been back on level terms when Flavio Paixao tried his luck with a shot from distance that Allan McGregor only just managed to tip over at full stretch in the Rangers goal.
At the other end, Steven Naismith saw his shot blocked in the packed goalmouth before the ball eventually broke for Lee McCulloch to rifle wide of target.
Rangers should have doubled their advantage after the restart when Steven Davis pounced on a slack pass before finding himself one on one with Cerny.
But he fired straight at the Accies goalkeeper who was able to block before eventually gathering Miller's attempts to follow up.
Kris Boyd's frustrating wait for his 100th SPL goal for Rangers continued when he was replaced by Kyle Lafferty, having done little to trouble the Hamilton defence on this occasion.
Joel Thomas had replaced Hamilton's top goalscorer Antoine-Curier but his attempt to restore parity was weak and comfortably smothered by McGregor.
Miller then squandered a great chance to put a comeback beyond their visitors when he collected a Davis pass only to balloon over the crossbar in front of goal when it looked easier to leave the net bulging.
Accies refused to give up hope of an equaliser and James McArthur - who confirmed earlier in the day he is set for a trial at Wigan later this month - had the ball in the back of the net but the effort was flagged for offside as Rangers held on for the win.