Home coach Lynn Howells had been forced to make a late change to his line-up - stand-off Hodge being called up in place of Phil Godman, who was laid low with a throat infection.
The Welshmen had the better of the nervy opening exchanges, but they were unable to turn their territorial superiority into an instant points advantage.
It was Edinburgh who edged in front in the fifth minute with a penalty by Paterson for offside.
The Scarlets hit back almost immediately with an identical score from Wales international playmaker Stephen Jones when the hosts were guilty of killing the ball at a ruck.
This exchange of kicks set the tone for the remainder of the half, with neither team looking likely to create a try.
Edinburgh sneaked in front again in the 23rd minute in the wake of midfield obstruction - only for Jones to find the target twice in quick succession.
It was a frustrating time for the Gunners fans because both strikes were set up by basic set-piece mistakes, the first at a restart kick and the second when they made a hash of their own line-out in the danger zone.
Hodge burst into the contest with a jinking run in the build-up to the interval and when the Scarlets were forced to handle on the ground, up stepped Paterson to complete his penalty treble, tying the scores in the process.
The boot went on to dominate the tussle after the restart as Edinburgh took a firm grip, making the most of Llanelli's increasing lack of patience.
Paterson confidently sent the ball between the uprights in the 42nd minute when the Scarlets re-offended at a ruck.
And he repeated the dose as the Welshmen were again culpable of crossing in front of a would-be tackler.
The match took a mysterious twist when Hodge took over the kicking duties.
But the outcome was the same as he fired an angled penalty over from at least 40 metres to stretch the Edinburgh lead to nine points.
He completed his double before Llanelli gave themselves a glimmer of hope with a Darren Daniel try.
But Hodge made sure of the home success with a slick drop-goal.
The scorers:
For Edinburgh:
Pens: Paterson 5, Hodge 2
Drop goal: Hodge
For the Scarlets:
Try: Daniel
Pens: Jones 3
Edinburgh: 15 Hugo Southwell, 14 Chris Paterson (captain), 13 Marcus Di Rollo, 12 Matt Dey, 11 Ben Cairns, 10 Phil Godman, 9 Mike Blair, 8 Dave Callam, 7 Alan MacDonald, 6 Simon Taylor, 5 Scott Murray, 4 Matt Mustchin, 3 Alasdair Dickinson, 2 Dougie Hall, 1 Allan Jacobsen.
Replacements: 16 Steven Lawrie, 17 Geoff Cross, 18 Chris Bentley, 19 Alasdair Strokosch, 20 John Senio, 21 Duncan Hodge, 22 Lucio Lopez Fleming.
Llanelli Scarlets: 15 Garan Evans, 14 Dafydd James, 13 Matthew Watkins, 12 Gavin Evans, 11 Darren Daniel, 10 Stephen Jones, 9 Clive Stuart-Smith, 8 Alix Popham, 7 James Bater, 6 Simon Easterby (capt), 5 Scott MacLeod, 4 Adam Jones, 3 John Davies, 2 Ken Owens, 1 Deacon Manu.
Replacements: 16 Craig Dunlea, 17 Aled Gravelle, 18 Lou Reed, 19 Gavin Thomas, 20 Liam Davies, 21 Ceiron Thomas 22 Morgan Stoddart.
Referee: Alan Lewis (Ireland)
Touch judges: Neil Paterson, Alan Forrest (both Scotland)