The red-hot Zurich Premiership front-runners take a seven-point lead into next weekend's return match and are hot favourites to advance to a quarter-final meeting against either Grenoble or Beziers.
Centre Andrew Higgins scored two second-half tries to help overcome a 12-18 deficit at the break, with the in-form boot of Ollie Barkley supplying 12 crucial points.
Colomiers hit the ground running with early tries to centre Jean-Phillippe Viard and winger Julien Arias, but five-pointers from Bath's Scottish international Simon Danielli and England scrum-half Martyn Wood keep the visitors in touch in a free-flowing match in front of 4,500 fans.
French international Thomas Castaignede was in inspired form for Saracens at Vicarage Road, scoring 20 of his side's 25-6 lead at halftime against an out-classed Glasgow outfit.
Castaignede scored two tries and added 12 points with the boot as the English club strode out to a 37-6 first-leg victory that makes next weekend's trip to Hughenden academic.
England scrum-half Kyran Bracken, Scotland hooker Robbie Russell and backrower Ben Russell added tries, while the Scottish province had to settle for two Dan Parks penalty goals.
At Kingston Park, the Jonny Wilkinson-less Newcastle Falcons ground out a tough 10-3 victory over Montferrand and face a daunting trip to France to make the quarter-finals.
England Sevens star Ben Gollings was the hero for the 5,000 home fans, scoring a first-half try, converting it and adding a second-half penalty goal to keep his side ahead.
Montferrand fly-half Jerome Gendre's penalty goal was his side's only reward for 80 minutes of hard graft, but the international-laden French club will rate their chances of overturning the seven-point deficit at Parc des Sports Marcel-Michelin next weekend.
London Irish secured a 20-13 win over Narbonne at Madejski Stadium, but won't be looking forward to next weekend's trip across the Channel to play the powerful French club.
Fly-half Mark Mapletoft slotted five penalty goals and left winger Justin Bishop scored his side's only try of a match that was held up for 15 minutes after an injury to his wing-mate Paul Sackey.
Narbonne winger Julien Candolon scored his team's try and fly-half Cedric Rosalen kicked eight points to give the French club a royal chance of advancing to a Challenge Cup quarter-final against either Connacht or Pau.
Elsewhere, visitors Beziers pipped Grenoble 19-15 and take a narrow four-point lead into a home match next weekend. Grenoble's inability to score a try proved costly, and a five-pointer to 30-year-old former French centre Jean-Marc Aue proved the difference between two closely matched sides.
Brive beat Castres Olympique 30-17 at Parc Municipal des Sports, with fullback Jerome Carrat kicking 15 points to complement tries to scrum-half Jean-Baptiste, winger Yves Donguy and former Cardiff lock John Tait.
The visitors had to make do with four Richard Dourthe penalty goals and a late try to replacement backrower Romain Froment, but will rate their chances of overturning the 13-point deficit next weekend.
Results: Brive 30 Castres Olympique 17, Colomiers 25 Bath 32, Grenoble 15 Beziers 19, London Irish 20 Narbonne 13, Newcastle Falcons 10 Montferrand 3, Saracens 37 Glasgow 6.