Nick Scott's interception looked to have given the home side a half-time lead against the run of play, but a brilliant quick-fire double from Tom May put paid to that and Luther Burrell's second half brace took the game beyond a relentless home effort that wouldn't cease, despite their eventual relegation to the Championship.
Stephen Myler gave Saints the perfect start, landing a pair of penalties inside the opening five minutes courtesy of his fine hanging kick off and a carving run from half-back partner Lee Dickson after a quick tap respectively.
With the returning Ken Pisi looking bright on the right Saints continued to threaten in attack but a couple of untidy set pieces would let Welsh off the hook and keep the sides separated by just a score.
Samu Manoa was next to go close, charging down Gordon Ross' kick but neither he nor Tom Wood could beat the Scotsman to the rebound and the Exiles bundled the ball clear again.
The visitors stuck with it but after another siege on the Welsh line ended in a turnover it was the men in red who would be frustrated, Saints scrambling just enough men back to eventually kill off the counter some 80 metres upfield.
All the game needed for its first try was for a pass to stick when it mattered, but unfortunately for the Saints when Dickson's pass did just that it was in the arms of the advancing Sonny Parker. Ben Foden showed electric pace to get back with the flying centre but Nick Scott was next on the scene in support and cruised under the sticks for the perfect sucker punch.
It wasn't long before Myler took Saints back ahead, just four minutes to be exact, and only three more before the visitors added a long overdue try. Patience paid off and having made telling inroads earlier in the move, May gassed the remaining Welsh defenders for a well worked score.
If that five-pointer was much needed, hot on its heels came a genuine contender for try of the season. The score was crafted by a stunning combination between Ken and Manoa straight from the restart, the first offloading and the latter surging round the outside of red defenders. With a huge overlap created Saints couldn't fail to score and May again sliced through a wide open Welsh backline.
That made the score 7-19 at the break and Saints almost produced an immediate continuation of the momentum, Jamie Elliott blasting up the left and only an unfortunate bounce quelled what looked likely to be a Saints third. Things were looking a lot more open as Welsh then found space, Foden and Ken scrambled the attack into touch five metres short - an intervention that would prove vital.
Moments later Saints had swept upfield and after bouncing off his opposite number Brian Mujati brilliantly laid on a Burrell try. Myler again cursed his luck as a second conversion attempt of the afternoon thumped the upright but Saints were fast sailing towards clear waters.
Welsh weren't going to cough up the points that easily, however, and continued to roll the dice from deep inside their own half. While their efforts were commendable, they were also causing Saints little worry, but when Alex Waller saw yellow for a late tackle the Exiles found a chink of light through the scrum instead. A man light and five metres from their own line, Saints were powerless to keep the barrage at bay and Daniel Browne was credited with a score that dragged Welsh within 10 and suggested the result might not be as cut and dried as it seemed.
As with the first home try, however, Saints quickly hit back and only a yellow card offence from Ross and a forward pass to Jamie Elliott prevented a fourth try. With Waller back on the park though Saints now had the man advantage and when the ball went wide, for the second time Burrell proved too hot to handle on the end of Wood's pass.
The centre's second finally killed off the hosts, ensuring another sizeable travelling contingent in black, green and gold went home happy and knowing that victory over Sale could be enough to secure a fourth straight Premiership semi.
2mins SAINTS Pen Myler 0-3
5mins SAINTS Pen Myler 0-6
30mins L WELSH Try Scott Con Ross
34mins SAINTS Pen Myler 7-9
37mins SAINTS Try May 7-14
39mins SAINTS Try May 7-19
HALF-TIME L WELSH 7 SAINTS 19
46mins SAINTS Try Burrell 7-24
64mins SAINTS Yellow Waller
70mins L WELSH Try Browne Con Ross 14-24
78mins SAINTS Try Burrell Con Lamb 14-31
FULL-TIME L WELSH 14 SAINTS 31