Jordan Hugill took full advantage of a badly weakened Swansea team to grab his first FA Cup goals for six years.
The 27-year-old striker, on loan from West Ham, was at Port Vale when he last got on the scoresheet in the famous old competition.
But a double before half-time helped fire QPR into Monday’s fourth-round draw, and manager Mark Warburton to a first ever FA Cup win.
Swansea boss Steve Cooper may have promotion from the Championship and a derby next week against fierce rivals Cardiff to worry about.
But in truth this tie was decided the moment the team-sheets appeared, with the Swans – quarter-finalists for the last two years - showing 10 changes from the New Year’s Day win over Charlton.
Rangers would have claimed unwanted history had they lost – equalling Plymouth’s record of 50 third-round losses. But that never looked likely.
Two superbly-taken goals from Hugill and another from the impressive Bright Osayi-Samuel made it 3-0 at half-time.
And after George Byers briefly gave Swansea hope, Lee Wallace - with a first for the club - and Josh Scowen extinguished it with wonderful late goals.
Connor Roberts was the sole survivor from the Swans’ last game, and Rangers’ line-up always looked stronger.
Warburton did hand a debut to 21-year-old Irish and ex-Liverpool centre-back Conor Masterson.
And soon QPR’s Moroccan winger Ilias Chair was turning the Swans’ defence inside out – but no one was there in the six-yard box to finish it off.
Goalkeeper Liam Kelly tipped a drive from Swansea’s Kristoffer Peterson on to and over the bar.
And the experienced Nathan Dyer, Swansea captain for the day, briefly threatened down the right. But the home side took the lead after 21 minutes with a goal of real quality.
Todd Kane surged down the right before putting in a cross that Hugill had no right to bring under control.
But the striker expertly caught the ball on his chest before volleying into the corner from 20 yards.
Osayi-Samuel, fresh from his two New Year’s Day goals against Cardiff, then robbed former QPR defender Jake Bidwell after 29 minutes and coolly found the bottom corner.
And on half-time Hugill grabbed his second from a Chair free-kick after Bidwell had fouled the on-loan striker.
Cooper made a triple change after 57 minutes – and almost immediately substitute George Byers pulled a goal back on the hour, sweeping home from 20 yards. Byers also hit a post late on.
Kristoffer Nordfeldt made good saves from QPR substitute Olamide Shodipo and Hugill to keep his side’s faint hopes alive.
Rangers made sure of a safe passage with another sumptuous volley after 76 minutes, this time from left-back Lee Wallace from the edge of the area. And Scowen completed the rout in style.