The black, green and gold were missing plenty of players for the match, and they lost many more during it.
But Saints stuck together and they won the game thanks to James Grayson's nerveless penalty from close to the halfway line three minutes from time.
The bonus-point victory was their fifth in succession in the Gallagher Premiership and they have now made their position in the top four a little more comfortable with two games to go.
"It was a pretty courageous effort, particularly in the pack," Boyd said. "We lost three guys before half-time and weren't able to freshen that pack up at all because we had to keep Fishy (James Fish) in reserve because we didn't want to go down to 14 men.
"It was a gutsy effort.
"We've been ticking away quite nicely while we've been playing knockout rugby and it was a pretty decent show.
"At the end I thought the roof was going to come off when we managed to get the penalty.
"Despite everything that happened, there were so many gutsy performances and I'm so pleased for everyone in the organisation, all the supporters and the long-suffering people.
"To keep ourselves alive and keep the roll going has been very pleasing."
Saints were 26-17 up at half-time before Quins hit back with a couple of tries after the break to lead 31-26.
But Grayson held his nerve with two penalties, providing the sweetest of wins for his side.
"We were pretty good for our money in the first 40 minutes and when we created stuff we managed to finish it," Boyd said.
"It was a mixture of good individual play, some good skill and also some really good team stuff.
"I'm proud because I was concerned with 20 minutes to go that we didn't have enough cavalry and we were going to run out of juice.
"We had to go deep in the well - and we did.
"It was probably the gutsiest performance we've had for a long time."