Manset scored the winner in the fourth minute of second-half injury time, after Jobi McAnuff and Adam Le Fondre had both found the net for the visitors in the final quarter to lay the foundations for an astonishing victory.
Albert Adomah and Brett Pitman had put the home side two goals to the good after an hour, but the visitors had a lifeline with 17 minutes left through McAnuff.
Two minutes later Le Fondre levelled from close range and a pulsating contest was won with 60 seconds of adding time remaining by Manset's cheeky flick.
Adomah had the ball in the back of the net after three minutes, racing clear of a static Reading backline to slip beneath the onrushing Adam Federici, but the Ghanaian's celebrations were cut short by a hugely debatable offside flag.
But, that aside, the opening 15 minutes lacked culture or quality.
That was until Adomah sprung Reading's lethargic offside trap once again on 23 minutes.
The winger latched on to Brett Pitman's through ball, outsprinted Joseph Mills and beat Federici with a slightly deflected effort.
David James prevented the visitors from immediately finding their way back into the match less than a minute later, saving smartly low to his left from Le Fondre's 15-yard volley.
Another tetchy period followed, before Maynard had a glorious chance to double the hosts' lead on 36 minutes but the forward could only hit the woodwork from 12 yards out, to leave the Robins leading by a single goal at the interval.
Seven minutes after the break, City had another glorious chance to put daylight between themselves and their guests.
A long ball was nodded by Reading defender James Pearce into the path of Maynard, who skipped past Federici but could not collect the ball with the goal gaping.
The Royals cantered down the other end of the pitch and Simon Church saw his header cleared off the line by Ryan McGivern, as the visitors began to find their feet. But out of nowhere find themselves further behind.
Neil Kilkenny's aimless lob forward seemed to be easy for Shaun Cummings to mop up, but he was outpaced by Adomah down the left who cut back for an unmarked Pitman to slot home.
Suddenly City were rampant. Pitman nodded straight at Federici, the Aussie goalkeeper produced top drawer saves to keep out McGivern and Adomah and Pitman struck the bar with a tremendous strike from the edge of the area. It was one-way traffic.
Reading reversed the flow on 73 minutes, when McAnuff fired home from 12 yards after James spilled Jay Tabb's cross.
And they turned the game on its head two minutes later, when James parried Hal Robson-Kanu's effort and Le Fondre followed up to tap home.
Neither side settled for a point, however, and Manset popped up with seconds remaining to tap a sneaky effort past the despairing James from McAnuff's centre to claim all three for the Royals.