Murphy, playing at fullback, with regular No.15 Tim Stimpson resting on the bench, scored all his side's first-half points through two tries and two conversions, which put Leicester ahead by 14-0.
Murphy's second try came with his captain Martin Johnson in the sin bin after a professional foul in the 31st minute. Johnson, who will lead England in the upcoming RBS Six Nations, now faces a one-match Premiership ban for being sin-binned three times in the last 12 months.
Bristol, who were man up, tried gamely to chip away at the early deficit, with Argentine playmaker Felipe Contepomi converting two shots at goal to put his side in with a chance at half-time, with the scoreline at 14-6 in the home team's favour.
But Leicester who welcomed Johnson back early in the second half, were the first side on the scoreboard in the second stanza when New Zealand flanker Josh Kronfeld crashed over for a try from an attacking line-out in the 50th minute. Murphy missed the conversion, but at 19-6 the defending champions had enough breathing space.
The last 10 minutes saw a flood of tries from Leicester, with outside centre Oliver Smith, his midfield partner Rod Kafer, and replacement fly-half Steve Booth all getting over for five-pointers.
Smith's try was set-up by Kafer, while Kafer's score came after a good break from No.10 Sam Vesty, before Vesty's replacement at fly-half, Booth, scored a good solo try.
By that stage Stimpson had found his way onto the field, adding all three conversions to take his team's final tally to 40, with Bristol failing to add to their six points in the second half.
The Tigers move into third place on the ZP standings courtesy of this win, with the Shoguns in 10th place, just one ahead of Newcastle Falcons.
For Leicester Tigers:Tries: Murphy 2, Kronfeld, Smith, Kafer, BoothCons: Murphy 2, Stimpson 3
For Bristol Shoguns:Pens: Contepomi 2