The black, green and gold snatched a dramatic win thanks to Taqele Naiyaravoro's try two minutes from time.
It came despite Saints having been reduced to 13 men following a red card for Tom Collins and a yellow for Cobus Reinach.
Saints had surrendered a 28-10 lead in the process, with Wasps taking a grip of the game at 31-28 up.
But Boyd's side kept battling and they eventually sent Naiyaravoro over the line.
They then kept possession for two minutes after the restart to make sure of a vital maximum haul in the Gallagher Premiership.
"The critical thing was that they didn't have the ball (after going 31-28 up)," Boyd said.
"At 28-10 up we started kicking the ball to them and saying 'you fellas can have a go now and see how you go', and they went bang, bang, bang.
"If they'd have managed to get the ball for one of three occasions in that last five minutes we might have been in trouble, but we did really well to keep the ball, keep the pressure on.
"They were trying to get the ball back but they infringed a few times and we managed to keep our composure and send the big fella over in the corner."
Saints had never previously won a Premiership game at the Ricoh Arena and had only won one of their past eight matches against Wasps.
But they set the record straight this time round.
And Boyd said: "We'd have been really frustrated and disappointed if we'd have lost.
"We were 28-10 up and seemingly in control of the game but we decided we didn't want to play any more so we shut up shop, which was disappointing.
"Then there was the aerial confrontation and thankfully both players were okay because Tommy (Collins) went down like a sack of spuds and unfortunately got a red.
"Cobus got a yellow and a penalty try so from 28-10 up we were suddenly looking down the barrel at 31-28 down.
"At that stage, we were sitting there wondering how we allowed that to happen.
"But there was good composure from the boys to organise themselves around what they needed to do to get some field position to score the try.
"There was a lot of discussion around whether we should have taken the penalty to level it up or to go for the lineout or the scrum.
"But at the end of the day, if you look at it from an option, execution, outcome point of view, the most important thing is the outcome and we managed to get our way over the top and then hold on to the kick-off and grind down the last two minutes to get five points.
"It was not necessarily a deserved five points but very crucial."