Ferguson, whose brother Derek starred for Accies under then-manager Allan Maitland, struck in the 35th minute with a crisp shot into the bottom right-hand corner after Brian Easton had only partially cleared a Steven Davis corner.
The Ibrox side deserved their win and could have stretched their lead on a few occasions.
Kyle Lafferty was first up inside the opening two minutes when his blistering 25-yard effort was parried by Tomas Cerny before being hacked clear.
Good work by the tireless Kenny Miller down the left saw him create space to cross for Davis, but his angled header flashed just wide of the right post in five minutes with Cerny struggling to meet it.
A moment of magic by James McCarthy almost saw Hamilton take the lead in the 13th minute. To a chorus of jeers from the Rangers end, the young midfielder flicked the ball up for himself on the right by-line before blasting a volley that Allan McGregor did well to turn past for a corner.
Three minutes later Miller twisted and turned past Chris Swailes before his rising shot was punched over the top right corner by Cerny.
Kris Boyd then had a shot well blocked by Accies captain Alex Neil, before Easton sent an effort just over at the other end.
But after Boyd was somehow let off with a scything challenge on Paul McGowan, Lafferty broke down the left and won a corner.
Davis' set-piece bounced around the box until Easton headed clear, but only as far as Ferguson who ripped a low shot into the net.
In the 38th minute Trent McClenahan was badly caught in possession by Lafferty on the right, who should have done a lot better than fire well past the left post.
Rangers nearly increased their lead in 42 minutes when Davis' corner from the right was nodded on by Kirk Broadfoot to Miller, but Cerny brilliantly pushed his effort onto the crossbar and over.
The second half was played at a much slower pace, with fewer chances. Boyd threatened in the 61st minute when he raced onto a Miller throughball, but his low effort was well held by Cerny.
Ten minutes later sub David Elebert was penalised for a foul on Boyd, but Cerny comfortably held Davis' effort.
Gers sub Steven Whittaker found space down the left before picking out Boyd at the back post four minutes later, but the striker stabbed his effort inches past.
Neil sliced a shot over as Accies tried to force their way back into the match, but it was substitute striker Kenny Deuchar who prevented the visitors from going ahead.
Davis was in acres of space on the right and had lined up a volley when the big striker threw himself at the ball to deflect it past for a corner.