Winger Flynn broke the deadlock in the 65th minute and right-back Brayford's deflected shot less than two minutes later ensured a deserved 2-0 victory for the resurgent Blades over Championship strugglers Charlton.
Nigel Clough's side become the first team from the third tier to reach the last four of the competition since Wycombe in 2001 and extend their remarkable winning sequence to nine matches in all competitions.
The Blades were spiralling towards relegation when former Derby boss Clough took over in November, but five months later the club are daring to believe a late run on the play-offs is on, as well as now securing their place in the FA Cup semi-finals for the first time in 11 years.
United had swept aside Aston Villa, Fulham and Nottingham Forest in previous rounds and save for a competitive first period, a determined Charlton could do little to stem the red and white tide.
Clough made seven changes to his starting line-up from the side that beat Peterborough in midweek.
Charlton striker Marcus Tudgay, in for Astrit Ajdarevic, was Charlton's only change.
The Blades started full of purpose, as expected after extending their winning streak on Tuesday night, and despite the changes looked anything but disjointed in the opening stages.
But Charlton made clear their own intentions in front of a sold-out Bramall Lane crowd of 30,040 - swelled by 5,000 Addicks fans - as both sides ensured a helter-skelter start without creating clear chances.
Tudgay's shot following Simon Church's lay-off was turned away by home goalkeeper Mark Howard and at the other end Blades defender Harris drove inches over the crossbar shortly before the break after Murphy had played the ball into his path.
Blades' impressive centre-half Maguire turned sharply in the opposition's box to fire the first effort of the second half just wide and Coady was inches away from connecting with Flynn's fine through-ball.
Charlton then spurned the best chance of the game up to that point. Tudgay did well to hook the ball back from the by-line, but Callum Harriott, who scored against Sheffield Wednesday in the previous round, dragged his shot wide from 10 yards.
It proved a costly miss as the Blades struck twice in as many minutes to seize control of the tie.
Flynn forced home the opening goal from six yards in the 65th minute with the sole of his right boot after Jose Baxter's curling cross from the left bounced into his path.
And before the home fans could catch breath, Brayford's low drive from the edge of the area took a deflection to wrong-foot Charlton goalkeeper Ben Hamer and nestle in his bottom left-hand corner.
Coady thought he had added a third with 13 minutes left after the ball rebounded back of the post, but his effort was ruled out for offside.
Charlton substitute Reza Ghoochannejhad went close in the closing stages, but Blades keeper Howard turned away his shot in style to ensure a sixth consecutive clean sheet for the home side and their place in the last four of the cup.
- Sports & Recreation
- Nigel Clough
- Marcus Tudgay