Paterson said: "We came together; we got what we are looking for. It was close - but we have done it.
"I think the conditions played a part. We knew we could win the game and we did - we are delighted to be in the quarter-final.
"We are burdened with the responsibility of 'progressing' the sport in our country. We have had a lot of hard times - this is the reward.
"We wanted to do it for everyone. We are not finished yet; we have a game next week that we are looking to win.
The Italians scored the game's only try - through Alessandro Troncon - and had 11 points from the boot of David Bortolussi.
But the latter missed a late penalty which would have sent the Azzurri through via this scrappy Pool C eliminator in St Etienne.
Scotland coach Frank Hadden paid tribute to the performance of his pack - and believes there is still plenty more to come from his side.
He said: "Once the rain came down, we knew it would be a tough day. I was delighted with the progress of our packs.
"We know how good the Italians are - I don't know how anyone can be dismissive of the quality of their pack. The way we held our nerve was a tribute to the quality of our side.
"We knew our defence would hold out. We had to make sure we made the right decisions when the opportunities came along.
"We are absolutely delighted that we have another opportunity to show the progress that we have made. We believe we can play better; now we have this one out of the way we can relax a bit more."