It was a thrilling opening to the new Gallagher Premiership season, with 11 tries in total, but it was Quins who emerged victorious thanks to two late scores from captain Alex Dombrandt and substitute Joe Marchant.
Falcons hooker George McGuigan opened the scoring before Quins responded through Andre Green and Wilco Louw. Adam Radwan and Nathan Earle then crossed to give the hosts a three-point half-time lead.
George Head and Lennox Anyanwu hit back for Quins before Newcastle thought they had snatched victory after McGuigan and Logo Mulipola registered, but it was Quins’ England pair who ultimately proved the difference.
Matson said: "We were on the ropes at times, but we created enough opportunities in the game. It will be the 31 points that we’ve conceded today that we will be our focus on Monday. They (Newcastle) have punished us when we made errors today, so you have to give them credit for that.
"One of the things I love coaching this team is that they are just so positive. They want to go to a scrum or to go for a lineout to score a try instead of taking the easier option. It’s one of those things that in the game, Newcastle were getting on top of us, but we did well to hit back and adapt to the opposition and the referee.
"The most pleasing thing about today is that we’ve taken five points in the first round, so it’s a stunning start and we are very happy to be getting on that plane home."
Newcastle boss Dave Walder chose to focus on the uplifting nature of his side’s performance, rather than the negative result.
Though Falcons finished second from bottom last season, they more than held their own against last year’s semi-finalists, leaving Walder upbeat after his first game in charge.
"I’m pleased and I’m disappointed today," said Walder. "I’m proud of the lads for the way in which they fought back into the game, but then also we couldn’t hold on. I think the way that the game ebbed and flowed we were never in complete control of it.
"They (Harlequins) play some great rugby and have a really exciting brand. You sometimes have to hold your hands up and say that that was good rugby.
"It will take a while for us to get to a level where it is the norm for the players to play an exciting brand of rugby, but it’s a start today. It’s a mindset change, but the attitude from the players has been outstanding."
Additional reporting from PA