Should the Falcons progress through that match they will then face a home semi-final against the winner of London Irish and Bayonne, who meet at the Madejski Stadium in the quarters.
Any potential semi-final would take place on the weekend of April 21-22-23 at Kingston Park, with the Falcons having won all six pool matches so far, including Saturdays 24-17 victory against Brive in southern France.
Director of rugby, Rob Andrew, said on hearing the draw: Connacht are a decent side, and we will not be taking them lightly at all in the quarter finals.
Our victory this weekend means we will get a home semi-final should we progress, but we are not even thinking about that at this stage, and we are focusing all of our attention in to what will be a very tough Guinness Premiership match at home to Saracens this Saturday.
Andrews men completed a lengthy journey back to the north-east on Sunday, reporting no major knocks from Saturdays win.