There were also tries for flanker Matt Gilbert, wing Liam Williams and replacement fly-half Jordan Williams, while Leicester could only manage a lone penalty from George Ford.
Scarlets had almost all the possession and territory in the opening minutes as they went through the phases, twice getting into the Tigers 22, but Leicester were the first to put points on the board when Ford kicked his penalty after 10 minutes.
The home side continued to play in an attacking style, with breaks from Dan Evans and Adam Warren setting up a try for Murphy who was sent over by fly-half Aled Thomas.
Almost immediately the Tigers had a chance to regain the lead. They were awarded a penalty right in front of the posts but chose to run instead of kick and knocked on.
Ford missed a penalty when Leicester chose to kick from longer range.
The Scarlets had a golden opportunity for a second try after 22 minutes when a perfect pass from Thomas put Rhodri Gomer-Davies through, but as the attack entered the 22 play was stopped for a forward pass.
Two tries then arrived in the space of two minutes as the hosts took control of the match.
The hard-working Murphy went over from 30 metres out in the 33rd minute after pouncing on a loose ball, and his fellow back-row forward Matt Gilbert slipped a tackle to score as the home side led 17-3 at the break.
Leicester would have hoped to come back strongly in the second half but a recovery did not materialise.
They went over the whitewash after 50 minutes through wing Ed Rolston but the try was ruled out for a forward pass, and as Leicester kept looking for an elusive try they twice coughed up possession in the Scarlets 22.
The Scarlets had not seen much ball in the second period but when they did get in an attacking position they made it count with wing Liam Williams running in for the bonus-point try after 69 minutes.
And the emphatic Scarlets win was capped in the 80th minute by a brilliant individual effort from 18-year-old Jordan Williams.
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
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Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
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The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.