But Pool Three remains wide open, with both Bristol and French champions Stade Francais waiting to pounce on any Cardiff slip-up.
Head coach Young said: "This game was a potential banana skin for us. Although Quins were realistically out of the tournament, we knew they wouldn't roll over.
"We now have a massive game against Bristol coming up. We are good enough to win there - but if Bristol are allowed to dictate the pace of the game we will have problems.
"We know how difficult it will be to go there and get a result - there are no easy games. "All you can ask for going into the last game of the pool is that your destiny is in your own hands."
Tries in each half from centres Dafydd Hewitt and Tom Shanklin, plus 13 points from the boot of full-back Ben Blair gave Cardiff their vital victory.
Quins, though, battled every inch of the way and matched the home side on tries through touchdowns for centre Hal Luscombe and full-back Mike Brown.
Young added: "It wasn't a perfect performance by us, but it was never going to be.
"What pleased me was that we didn't panic and we managed to close the game down. I was pleased with our control and our mind-set.
"We are happy where we are in the group. We have now got to make sure we can get over one more big hurdle."
Quins rugby director Dean Richards saw his team once again come unstuck - they have recorded just one victory from their last 11 starts in all competitions - but he could feel satisfied with aspects of the display.
Richards said: "I thought there was a huge amount of endeavour.
"There were some very pleasing things. Our front row played really well and our scrum totally dominated their scrum.
"Game control is what we are lacking at the moment - the ability to turn the screw when it matters."