Ross Tokely earlier saw red for the Highlanders as Nikica Jelavic converted from the resultant penalty before Edu was also dismissed, having netted the rebound after Jelavic was denied with a second spot-kick.
The three points were a major boost for the Scottish champions at a tough venue but they were left facing a defensive crisis when Kyle Bartley - who last week agreed a second loan spell from Arsenal - limped out of the action in the first half.
With skipper David Weir still sidelined and Steven Whittaker out with a broken toe, Lee McCulloch was also added to the list of injuries after suffering a knee problem ahead of the match, with Rangers director of football Gordon Smith confirming in an interview with BBC Scotland before the game that they have now withdrawn their interest in Aston Villa defender Carlos Cuellar.
Bartley initially started in the heart of defence alongside Dorin Goian, who recovered from a hamstring injury, while Jamie Ness made his first SPL appearance of the season after being drafted into the midfield for the clash in the Highlands and Kirk Broadfoot also started.
Caley Thistle made one change from their draw against Dunfermline last week, with Graeme Shinnie replacing the injured Kenny Gillet.
Once the match was under way, Jelavic had the ball in the back of the net with just three minutes gone after pouncing on an Edu pass, but the offside flag was raised.
Rangers were handed a free-kick on the edge of the area when Jelavic was fouled by Tokely but Lee Wallace's effort was blocked by the wall, before the rebound from Ness was deflected wide.
Inverness had a decent chance of their own when Jonny Hayes embarked on an impressive run before setting up the shot for Aaron Doran, but his effort was weak and comfortably smothered by goalkeeper Allan McGregor.
Gregory Tade found himself one on one with McGregor only for the Rangers number one to claim possession before the Inverness forward could unleash the shot.
Rangers were dealt a major blow after 33 minutes when Bartley was forced out of the action following a challenge from Tade and was replaced by youngster Ross Perry.
Inverness could have snatched the opener just before the break but Richie Foran just failed to connect with a Doran cross in front of goal.
The Highlanders still looked dangerous after the restart and Tade was not too far away with a 25-yard drive that dipped just over the crossbar.
But the match swung in Rangers' favour when Tokely was dismissed for a challenge on Steven Naismith in the box as the referee pointed to the spot.
Jelavic stepped up for the penalty and kept his cool to convert the pressure kick and hand the visitors the advantage with 60 minutes gone.
Rangers were awarded another spot-kick, this time for a Shinnie foul on Goian. Jelavic was denied his second goal of the day by a decent save from goalkeeper Ryan Esson but Edu was first to react and rifle home the rebound after 68 minutes.
However, the American's joy turned to despair with 75 minutes gone when he was sent off after being shown a second yellow card for a challenge on Nick Ross.
Substitute Gregg Wylde - whose future at Ibrox has been the subject of much speculation - could have wrapped up the win but blasted well over late on.