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Norton-Knight, back in the Blues starting line-up for the first time since a disastrous debut in the season opener against Edinburgh, was still wayward with his kicking but linked up well with his centres and made a half-break that led to Blair being held up at the corner.
Scarlets full-back Daniel Evans slotted over a penalty to give the visitors the lead, but Blair cancelled it out before the Blues finally crafted an excellent try.
A touchline gallop by Richard Mustoe, stopped by Evans at the cost of a sore head, sparked a fine handling move that freed number eight Xavier Rush, whose inside pass saw fellow New Zealander Paul Tito crash over.
A third New Zealander, Blair, converted from wide out, and added a 50-yard penalty before half-time, but the Blues' joy was tempered with concern for Lions centre Jamie Roberts, who left the field after a bruising collision with Sean Lamont.
After Evans made a hash of a long penalty for the Scarlets, Blair extended the home side's lead with two further kicks as burly Andy Powell joined the fray to give the Blues' attacks some added impetus.
But after losing the talismanic Rush with a hand injury - which forced them to play a lock in the back row - the Blues found themselves under severe pressure in their own 22, returning Lion Matthew Rees being denied a touchdown after a prolonged examination by the TV official.
It merely delayed the inevitable, as Scarlets number eight David Lyons drove off a five-metre scrum and shoved his way through fellow Australian Norton-Knight to the line, with Evans adding the conversion.
A massive gap opened up in the Blues midfield, allowing fly-half Rhys Priestland to race through and lay on a try for centre Jonathan Davies, but Evans' missed kick left the Scarlets still needing another try to snatch victory.
It proved too much, and despite a lengthy siege under the Blues posts the visitors were unable to break through, and the home crowd, a healthy 12,198, were able to celebrate in relief.