The black, green and gold scored four tries in an incredible final 15 minutes as they roared back from 31-12 down to win 36-31.
The bonus-point success moved Saints into the Gallagher Premiership top four, having played the same number of games as the sides they leapfrogged: Exeter and Gloucester.
And Saints boss Boyd was delighted with how his side responded after a badly below-par first 65 minutes.
"I thought we were terrible for the first 60 minutes," Boyd said. "It's the worst we've played for 60 minutes for a long time.
"Probably at the point that Brandon (Nansen) went down (with an injury) we managed to rally, and we snuck over in the end.
"We're pleased to take the five points and get the Rec bogey off our back.
"It was a horrible first 60 minutes but we're very proud of the character shown by the guys to come back and score those tries at the end to win.
"The language was a little bit blue at half-time.
"We'd created very few chances in the first half, and the ones we created we butchered.
"We weren't creating anything because we weren't winning any collisions and they were on the top of us.
"I thought when we came back and scored a try after half-time, we could settle down, but they scored a couple and it went out to 31-12 at one stage.
"At that stage, I thought the season was over and we were going to be playing for pride, but we're very pleased to get the victory.
"The boys know they've used their get-out-of-jail card and we won't get that opportunity against Harlequins on Friday or Saracens next month.
"We know we have to be way better than that but we had a very disrupted week with Dan Biggar and Courtney (Lawes) being out.
"Dave Ribbans tried to get through the week and didn't make it, Fraser Dingwall tried to get through the week and didn't make it so it was a bit of a disrupted week.
"We go into a short week now that is going to be a bit disrupted as well so we've got to fight through that."