Prop Carre was shown a straight red card for a shoulder-led tackle on Dan Evans just four minutes in, but the reigning champion went on to claim a crucial 22-15 triumph at Liberty Stadium on Sunday (AEDT).
Five-eighth Manu Vunipola was successful with his six efforts from the tee, which consisted of five penalties and the conversion of Alex Lewington's try off Richard Wigglesworth's box kick.
Sarries consequently moved up to second in Pool 4 and three points behind Racing 92, which faces Munster on Monday (AEDT).
A battling 31-31 draw with Glasgow Warriors was sufficient for Exeter to reach the last eight for just the second time in its history, with only the crossbar denying it the win via a 60-yard Stuart Hogg penalty beyond the 80th minute against his former club.
Tommy Seymour touched down in the opening minute as the Warriors surged into an early 14-3 lead, but Matt Kvesic's double helped ensure the Chiefs got a bonus point and progressed as Pool 2 winners.
Second-half tries from Alivereti Raka and George Moala led Clermont from behind to a 29-13 win at home to Ulster, the Top 14 side taking control of Pool 3.
Toulouse guaranteed a last-eight berth by winning 21-7 at Connacht. Jerome Kaino, Julien Marchand and Pita Ahki cross for the visitors, which were reduced to 13 men in the closing stages when Zack Holmes saw red for a dangerous tackle after Yoann Huget was sent to the sin bin.
Gloucester lost Danny Cipriani to injury after 12 minutes but remained in contention to advance as one of the three best runners-up with a 29-6 bonus-point win against Montpellier.
Ben Morgan, Louis Rees-Zammit and Todd Gleave marked their returns to the starting line-up with tries, the latter coming after Paul Willemse was shown a yellow card for collapsing the maul.