Referee Matthew Carley, taking charge of his first Premiership match, issued five yellow cards and could easily have sent off Welsh's Russian lock Kirill Kulemin for a punch on Worcester's Chris Jones.
But a try bonus point victory was deserved for Welsh who got tries through wing Dan Caprice, number eight Alfie To'oala, full-back Tom Arscott and wing Nick Scott, with fly-half Gordon Ross adding 13 points with three penalties and two conversions.
It was Welsh's first league win since December 1 and only the fifth of what has been a fraught debut season in English rugby's top flight.
Worcester only fired after the break with two tries from Josh Drauniniu and one from replacement back row Jake Abbott plus a penalty and two conversions from stand-off Andy Goode.
The Exiles, who were docked five Premiership points for fielding an ineligible player this term, were condemned to bottom place in the table last month but looked eager to go out with a bang as Worcester looked to be planning their end-of-season holidays with a dreadful first-half performance.
Indeed Welsh only had 14 men on the pitch for 10 minutes of the half after Kulemin, number eight Dan Browne and Ross found themselves in the sin-bin. Worcester were guilty of a yellow card as well as flanker Neil Best spent 10 minutes on the sidelines.
Welsh went at the game from the start and were 14-0 up with just nine minutes gone.
Caprice got the first of their two tries to that point when he raced towards the right corner from midfield 30 metres out, rounding the withering Worcester defence.
Immediately after Best saw yellow for a ruck offence, Welsh's pack drove 10 metres to the line for the excellent To'oala to cross.
Worcester, whose handling let them down, had a golden chance to reduce the arrears once they were back up to their full compliment and Welsh were a man down as a rolling maul put them over the home line but, inexplicably, they could not ground the ball.
It summed up a frustrating half for captain James Percival and his side which got worse when Ross booted a penalty before Goode finally got the Warriors on the board with three points.
It was looking all but over within 12 minutes of the re-start as Arscott raced through a huge gap in the defence from 10 metres before Scott scored a dazzling touchdown, a 75-metre run down the left, to secure the fourth try and the bonus point.
That made it 30-3 but Worcester bravely hit back when the opportunity arose through two tries in three minutes from flyer Drauniniu. He first intercepted a loose pass inside the Exiles' 22 and jogged over unopposed from 10 metres.
His second was from longer range as he sprinted down the middle to run in under the posts. Goode's boot added the extra points and suddenly Worcester were back in the mix at 30-17.
Yet the Warriors let it slip away and did not help themselves when lock Chris Jones became the fifth player in the bin.
A final flourish saw Abbott go over but it was too little, too late as Welsh bowed out of the Premiership with a win.