The home side made a strong start to the contest between the Pool One and Pool Four heavyweights, crossing over inside the opening 15 minutes through Tim Streather, who batted down a punt from the opposition and made the breakthrough.
Nick de Jager made things even better shortly after when he finished off a fine team move by crossing over in the corner, setting the tone for what was a convincing Sarries win in the end following a minor shaky spell.
Having been punished for not rolling away, Sweeney was once more granted the chance to take range, and he was on target again on the brink of half time to successfully convert the penalty and set up a tight contest in the process.
That proved to be the end of Exeter's scoring, though, and from that stage on it was all about the home side as they racked up three tries - all successfully converted by Ben Spencer - to claim victory.
De Jager blasted his way through for the first of that trio, while Catalin Fercu capitalised on Spencer's solid vision to add his name to the try list.
It was left to Matt Hankin to round off the scoring at Allianz Park, crossing over on the back of a well-executed driving maul a little over 10 minutes from time.
Saracens will now take on Northampton Saints in the semi-finals, while the defending champions travel to Leicester Tigers.