Saracens coach Mark McCall admitted the red card issued to Jared Payne was critical in their 17-15 victory over Ulster at Ravenhill on Saturday.
Ulster's full-back was sent off in the fourth minute for taking out opposite number Alex Goode in the air, with Jérôme Garcès showing him red.
Despite having a numerical advantage, Saracens went in at the break 5-9 down to the hosts, who also lost Rory Best and Ruan Pienaar to injury.
But tries from wing Chris Ashton (2) and lock Mouritz Botha saw Saracens home and they will now face Clermont at Twickenham in the semis.
"Ulster deserve an enormous amount of credit after losing a man after five minutes and, not only from the effort they put in, but to also be as tactically astute and aware as they were was a great achievement and we're just lucky we got over the line at the end," McCall said afterwards.
"It definitely wasn't intentional (Payne's collision) but it was reckless. You could argue both ways, but it's a big call to make five minutes into the match and it changes the match.
"At times we played some good rugby and at times our physicality was outstanding, but we kept on showing unbelievable indiscipline.
"We gave away too many penalties and we kept them in the game and we kept the crowd in the game, and we were fortunate to get away with the win."