The Warriors made it four bonus-point victories from four outings - and a perfect 20 points - with another impressive performance against Saracens, adding to earlier wins over Sale Sharks, Toulouse and Clermont Auvergne.
That run of results secures Franco Smith's side a home tie against the Vodacom Bulls in the Round of 16, as the Warriors continue to underline their credentials on the European stage.
"You dream and you hope you're going to get four wins from four and I think that's the difference this year - we actually believe that," said Horne, who marked his 150th appearance for the club with a try and a Player of the Match display.
"We've played two of the most successful teams in Europe over the past few years, and we've shown that if we play our best, we can match them."
Despite the perfect start, the scrum-half was quick to stress the importance of keeping focus as the season progresses.
"We've done our job over the first four games to put ourselves in the best place, but you can't look too far ahead or you'll come unstuck.
"So now we've got to focus on the league for the next few weeks and then the Bulls coming here.
"We've had a few tasty battles against them in the last couple of years, so we know that will be an absolute dogfight as well."