First-half tries from Ryan Grant and Tim Swinson put the Warriors in the driving seat.
The Scarlets battled back through the collective boot of Rhys Priestland and Jordan Williams but it was not enough as Glasgow maintained their unbeaten start to the season.
“They didn’t have to work for their 14 points at the start of the game and that was down to our lack of intensity,” said Easterby.
“We came out far stronger in the second-half, had dominance in the scrum and control over lots of things but we lacked composure.
“I can’t fault the effort and there were certain things that were pleasing but the result is the only thing that matters.”
It took in form Glasgow less than three minutes to open the scoring at Parc y Scarlets. Niko Matawalu set the Scarlets’ alarm bells ringing with a quick tap and following several pick and drives, Grant rumbled over.
Duncan Weir added the extras but Glasgow continually found themselves on the wrong side of Italian referee Giuseppe Vivarini and back-to-back penalties against Al Kellock saw Rhys Priestland cut the deficit.
But Glasgow grabbed a second try as Mark Bennett fielded Jordan Williams’ up and under and following a succession of neat off-loads Swinson twisted over.
Weir added the conversion but Priestland fired over a late penalty to enter half-time trailing 14-6.
The Welsh region struck with Priestland’s third penalty within moments of the restart and they should have quickly surged in front.
Following a mix-up between Sean Maitland and Niko Matawalu under the high-ball the door was opened for the Scarlets.
Jordan Williams pounced and despite selling a dummy to go through, he offloaded to Liam Williams, who was tackled by the Fijian flier.
The west Wales region dominated the majority of the second-half and Williams eventually cut the deficit to just two points.
But Weir quickly responded Glasgow clung on despite a late yellow card to Matawalu.
