The defeat means that Welsh no longer have any chance of reaching the semi-finals, while Saracens maintain their 100% record in the competition this season and now just need to win their final game to book their place in the last four.
© Getty Images
It was the hosts who were first to register on the scoreboard as Tristan Roberts sent a drop goal through the posts to give his side a 3-0 lead after 12 minutes.
Welsh almost claimed the first try of the afternoon as well, but George Perkins got across and made a crucial try-saving tackle to deny the hosts.
Sarries were struggling to get going in attack, relying largely on their defensive work to keep Welsh at bay, but it was the visitors who went into the break ahead as Ben Spencer collected a Max Wilkins kick to score right on the stroke of half time.
They began the second half in similar fashion, going over within two minutes of the restart as Jared Saunders burrowed his way in.
The hosts were quick to respond, however, and got their first try of the afternoon just three minutes later when Opeti Fonua barged his way over the line from a few metres out.
Things went from bad to worse for Sarries as they fell behind to Fonua's second try of the match before Josh Buggea was shown a yellow card to leave them with just 14 men.
They were handed a reprieve when Alex Davies fired a penalty wide on his return to action, and the visitors managed to prevent Welsh from scoring at all during Buggea's time off the field.
Spencer edged his side ahead with a penalty late on, and a second kick from inside his own half opened up a five-point gap in the closing stages of the match.
That proved to be enough for a victory that sees Saracens leapfrog Gloucester back to the top of the group.