A 19-14 win over Toulouse at Murrayfield saw the club reach the last four of Europe's elite club rugby competition for the first time.
Edinburgh Rugby will face Ulster in an unprecedented and historic Heineken Cup semi-final at the Aviva Stadium, Dublin, on Saturday, April 28.
Laidlaw, who captained the side, said: "The fans were immense today. The noise they created in the stadium was amazing, from start to finish, but towards the end of the game they were just incredible.
"It was an exhausting and extremely physical game and in the last 10-20 minutes they made a huge difference as we battled on for the win. They should all take pride in the part they played today. They were immense."
Scottish Rugby chief executive Mark Dodson, echoed the sentiment of the man of the match-winning captain. He said: "Edinburgh Rugby have earned a place at the top table of European rugby today and have given the nation a big boost with this incredible performance and victory.
"There was so much to this match today. The event was spectacular from start to finish and the players were part of a great show here and long may that continue." And he added: "All of Scottish rugby wish them all the best for the semi-final in Dublin."