Attendance 49,918
This was the first time since last May that the 'Gers had overcome the Tannadice men - the previous two fixtures this season having ended all square.
Manager Walter Smith made two changes from Hampden with Steven Whittaker and Fleck replacing Sasa Papac and Steven Naismith.
Both sides were cautious right from the opening minutes, each offering their opponents the maximum possible respect - but it was the visitors who were first to threaten in eighteen minutes when Garry Kenneth's long throw found David Robertson whose shot was held by Allan McGregor.
Six minutes later Rangers should have opened the scoring when Pedro Mendes found Nacho Novo down the right, a surging run ending with a square ball to the unmarked Kris Boyd who somehow curled his shot wide.
The home side were beginning to secure a grip on the game - both Boyd and Barry Ferguson had shots blocked following a Fleck free-kick on the half-hour, then almost immediately the teenager's 25-yard left-foot shot was touched over by visiting goalkeeper Lukasz Zaluska.
Rangers however suffered an injury blow in 42 minutes when Novo limped off with an ankle injury to be replaced by Kenny Miller. The little Spaniard had been limping for some time and it was a mystery why it took the Ibrox Management so long to substitute him.
The first-half ended goalless - but Dundee United had ruined the game as a spectacle with ridiculous time-wasting tactics that will kill football stone dead.
The spoiling tactics continued as the second-half developed - but an individual piece of brilliance from Madjid Bougherra in 64 minutes almost broke the deadlock when his surging run down the right ended with a cross to Boyd whose hook shot was held by Zaluska.
Nine minutes later the ineffective Boyd was replaced by Kyle Lafferty - and it was to prove the turning-point of the game.
Within sixty seconds Rangers had the ball in the net when Lafferty controlled a chip from his fellow Ulsterman Steve Davis before slotting the ball home only for the goal to be disallowed for offside on the say-so of linesman Alan Cunningham.
The opening goal was merely delayed - and it arrived via the penalty spot in 76 minutes after Fleck had been bundled to the ground by Mihael Kovacevic. It was a stonewall penalty that not even Tannadice Boss Craig Levein could dispute - and youngster Fleck coolly stepped forward to send Zaluska the wrong way from the spot.
Seven minutes later it should have been 2-0 when in a lightning break Davis released Miller whose first effort was blocked by Zaluska - the second flashing across the face of goal.
There was no time-wasting from the Tannadice men now - but it was Rangers who were in command, Davis seeing his 22-yard shot from a Kirk Broadfoot pass being held by Zaluska in 88 minutes.
With the seconds ticking away United's Polish goalkeeper denied Lafferty the clinching second when he blocked the Northern Ireland International from a Davis cross.
Referee Iain Brines somehow decided that there should be four minutes of injury time, to utter astonishment from the capacity crowd. If this was for United's time-wasting, then they were
enjoying a double advantage.
It mattered not one jot however - in the final seconds Lafferty calmly slotted the ball home into an unguarded goal after Ferguson, clean through on goal, had been blocked by the out-rushing Zaluska.
Rangers were worthy winners - and afterwards Walter Smith summarised:
"We worked hard for our win. We never really got started, but played well after opening the scoring. United were difficult to break down. John Fleck showed great confidence to take the penalty."
RANGERS McGregor; Whittaker, Weir, Bougherra, Broadfoot; Davis, Mendes, Ferguson, Fleck; Boyd (Lafferty 73), Novo (Miller 42)
UNUSED SUBS Naismith, Alexander, Papac, McCulloch, Aaron
DUNDEE UNITED Zaluska; Kovacevic, Dods, Kenneth, Grainger; Feeney, Robertson (Daly 79) , Gomis, Buaben, Conway (Swanson 56); Sandaza (Goodwillie 69)
UNUSED SUBS McGovern, Dillon, Shala, Dixon
REFEREE Iain Brines