The black, green and gold played more than a hour with 14 men after David Ribbans was sent off for shoulder contact to the head of Luke Cowan-Dickie.
Saints refused to relent during the first half, lifting their intensity levels through the Franklin's Gardens roof. They grabbed an 18-0 half-time lead, with centre Rory Hutchinson scoring twice.
But champions Exeter came roaring back after the break and they eventually earned a 29-26 win thanks to Joe Simmonds' penalty seven minutes from time.
But Boyd and the 4,000 fans inside the Gardens had every right to be proud of Saints' hugely spirited showing.
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The Saints boss said: "It was close but no cigar.
"At the end of the day, when we were trying to decide whether to try to kick the goal at the end, the draw, the win, the loss wasn't really important - what was important for us was the character, the resolve and the determination to work with 14 men.
"Exeter started winning those collisions in the second half so instead of being on the front foot and really slowing their ball down, we were getting penalised.
"They kicked to the corner, got their lineout drive going and got three tries.
"We know what we need to do when we play Exeter to get a performance and we did it for as long as we physically could.
"We just couldn't quite slow the ball down so we started infringing and they got up the field and came away with the win.
"It was a tremendous effort from our guys."
Saints, who finish their season at Bath next Saturday, will definitely end the campaign sitting fifth in the Gallagher Premiership.
And Boyd said: "It's a disappointment that we didn't put on a better show and get in the top four but the four teams that are there have all had good seasons and they are four worthy semi- finalists."