The black, green and gold go to Coventry on Sunday, when they will be bidding to make it four wins from four at the start of the new Gallagher Premiership season.
Saints squeezed past Irish last Saturday, earning a 23-21 success thanks to a Dan Biggar penalty six minutes from time.
But Boyd knows his team must up their game significantly if they are to see off a Wasps side who crushed Bristol Bears in their only home match of the season so far.
"We'll have to be massively better against Wasps, but we needed to be better than that against Irish," Boyd said.
"So we just can't afford to play like that because this competition is going to be massively close.
"That dogfight in the league table is going to be anyone's guess so it makes for a good season."
While Boyd has not been overjoyed with Saints' performances so far, he has been happy with the outcome of their matches.
They beat Gloucester and Exeter before edging to victory against Irish.
And Boyd said: "It is a pretty good start.
"The puritans in all of us would love us to play good rugby but it's high-performance sport after all, and I'd rather us be three wins from three and us playing very poorly or very poorly in patches, than be playing magnificent rugby and have lost three games on the bounce.
"We've just got to find a way to hold on to momentum and territory, and then when we come under pressure wrestle the initiative back again.
"We're on a bit of a path to get more bloody-minded around some of that.
"I suppose three from three to start is showing that.
"But we know that this week, the week after and the week after that, if we play like we did last Saturday we won't get that result."