With both clubs selecting strong line-ups, the hosts took full advantage of a sluggish Quins start to lead 15-3 at the break before the visitors staged a second-half revival.
Duncan Taylor, Mike Ellery and Jack Wilson touched down for Sarries as Ben Spencer kicked two penalties, while Mark Lambert and Charlie Walker crossed the line for Quins in the second half with Tim Swiel and Louis Grimoldby both adding three points.
And while the Premiership and the European Champions Cup may feature at the forefront of minds at Allianz Park, Shaw is adamant Saracens aim to be challenging for silverware come the business end of the season.
“Each competition is important to us from the A League up,” said Shaw. “Obviously you have to put youth in at certain stages but we want to win every game.
“We were in control but had two tries ruled out by the TMO. We tried to force things too much in the final quarter and let them grow into the game.
“However, Quins had a bit of experience in their line-up and a lot of talent so we are more than happy with the result.”
Despite being content with the resurgence of his side during the second period, Quins boss Conor O’Shea hinted that he might be tempted to lean towards greater experience for their next clash with Newport-Gwent Dragons on Friday.
“We showed great mental fortitude as we reacted to a massive challenge as the game went on,” said O’Shea.“They had six internationals in the pack and some serious performers. We may play a few more senior players against the Dragons as it would be nice to have something to play for in the LV=Cup after Christmas.”