The capital city club charged into the last eight of the competition at Murrayfield on Sunday night, with a 34-11 win over London Irish. The victory saw Edinburgh top the group and secure a home quarter-final against four-time champions, Toulouse, on Easter weekend.
Laidlaw said: "It's fantastic to make it this far but we're not finished yet."
He added: "There's a lot of belief and talent in this young team and we're determined to go further than any Scottish team has before and book a place in the semis in front of another record-breaking crowd."
Laidlaw was also keen to pay tribute to Edinburgh's supporters for the part they have played in the club's progress so far.
He added: "It's a tremendous personal achievement for the players and coaches to secure a home quarter-final and we're delighted to be able to repay the support of the fans who have been right there with us throughout the group stages at some incredible matches.
"Last night's game was the perfect example of the fans playing their part in another great win.
"We all knew that, no matter what, we needed that fourth try to get the bonus point and the noise they created as we piled on the pressure was awesome."
Munster v Ulster
Leinster v Cardiff Blues
Edinburgh v Toulouse
Saracens v Clermont Auvergne
Saracens/Clermont Auvergne v Leinster/Cardiff Blues
Munster/Ulster v Edinburgh/Toulouse