Scotland prop Allan Jacobsen crossed with Netani Talei and Ben Cairns following suit before James Downey reduced the deficit by the break.
Paul Diggin's second-half try and 16 points from the boot from Stephen Myler finally saw the Saints through.
Bath eased past Italian side Aironi 22-6 at the Stadio Luigi Zaffanella to end their run of four straight European away defeats.
Tries from Tom Biggs, Matt Carraro, Nick Abendanon and Luke Watson handed the English side victory, who desperately needed a win following their opening pool four defeat to Biarritz.
Ludovic Mercier managed two penalties for the hosts, but once Bath scored in the 11th minute of the contest, the result was never in doubt.
Saracens have it all to do to after they lost their second successive game - Ireland starlet Johnny Sexton scoring all of the Leinster's 25 points in their 25-23 win at Wembley.
Alex Goode replied with 17 points for Brendan Venter's men but it wasn't enough as Joe Schmidt's men did enough.