Laidlaw struck three conversions and five penalties at the Stadio Olimpico, scoring 21 points, as Scotland ended their miserable nine-match losing streak in the Six Nations.
It was their first win in the competition since February 2014, a first for coach Cotter and many members of his squad.
“It means absolutely everything to us,” Laidlaw told ITV, “to this jersey, to the players who have worn it in the past, and to the players who are in it now.
“We’re trying to build a culture here, a ‘no excuses’ mentality, and you’re starting to see that come through.
“We’ve won the game comfortably and I think it’s fair to say we deserved to win,” he added.
“For the young players, for Vern, for all the coaches, it’s a great time and we can’t wait to get back to Murrayfield, to wear the jersey again and to try and get a win against France.”
Scotland finished bottom last year but after their fine showing on Saturday, Cotter’s side have a platform to build on while Italy, who have now lost their last seven home matches in the competition, look favourites for the wooden spoon.
“It’s a bit of a relief, the team’s been improving and we can probably improve in a few areas as well. We won’t sit back and say that’s the finished article because it’s not," he said.
“We’ll just enjoy our week I think, and then we’ll look at things we can improve and keep on moving forward.”
(Reporting by Ed Dove, Editing by Ken Ferris)
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