With a wealth of attacking talent such as fullback Stuart Hogg and flyhalf Finn Russell, Scotland have proved they can score against some of the best sides in the world in recent years. Wins like Saturday's though have been few and far between.
"We are a changed group and we want to drive this whole thing forward," said Laidlaw, who landed two penalties in the final 10 minutes to secure Saturday's win.
"When we pull these jerseys on at home, we don't want to get beat anymore."
Scotland have a dismal Six Nations record, having finished in the bottom two 11 times in the 17 seasons since Italy joined the fold. They have not won on the opening weekend for more than a decade.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
Laidlaw said his team, coming off the back of another last-gasp win over Argentina in November, now had a plan for squeezing out victories.
"We know what plays to go to when in tight games. We know to keep tight balls... to keep pressure on teams... to grab a bit of field position," said Laidlaw.
Outgoing coach Vern Cotter has suffered some heartbreaking defeats during his tenure, including single-point losses to Australia last year and in a 2015 Rugby World Cup quarter-final.
He had an uncomfortable wait at Murrayfield on Saturday for what he said was his best win as Scotland coach.
"I was struggling up in the coaches' box towards the end. We lost a bit of purchase in they second half... but credit to Greig and the boys they got it back," said Cotter, who is leaving for French club Montpellier in the summer.
"A lot of things came off... we were physical at the breakdown and the guys pulled up their sleeves and were accurate."
Cotter's Irish counterpart Joe Schmidt said he had always feared an upset could be on the cards and also praised Scotland's new-found winning mentality.
"Those losses they (Scotland) have had have been very narrow margins, so they have always been there or thereabouts. A little bit of belief and confidence just starts to self-generate and make them play better.
"They probably are a little bit more canny, and they have a bit more confidence, so the fine margins start going their way."
(Editing by Clare Fallon)
