Sneak up without a Torch

  1. Put the Torch away before approaching.
  2. Crouch and move slowly toward the Rat.
  3. Catch the Rat.

The capture advice comes from the developer guidance collected in NPC Tips' Leather guide. If chasing is frustrating, use the passive trap option instead.

Rat inventory icon
Rat

Use a Rat Trap instead of chasing

The hand-crafting recipe is 3 Sticks → 1 Rat Trap. Set the trap and return later: it is the passive alternative to sneaking up on a moving Rat.

Keep your Torch for the route afterward, but put it away again when you switch back to manual capture.

Make Rat Trap
  1. Ingredients

    Stick inventory icon
    3 × Stick
  2. Craft at

    Hand crafting

  3. Makes

    Rat Trap inventory icon
    1 × Rat Trap

Process the Rat before making Leather

Use the hand-crafting Raw Meat recipe with 1 Rat. The same step produces 1 Animal Hide as well as the meat. Capturing a Rat is only the start; this processing step supplies the Hide.

Save 2 Animal Hide for 1 Leather at a Workbench. Two processed Rats cover that cost. You still need to pick up Leather once to unlock its recipe, so having the two Hides is not enough by itself.

Process a Rat: meat plus an extra Hide
  1. Ingredients

    Rat inventory icon
    1 × Rat
  2. Craft at

    Hand crafting

  3. Makes

    Raw Meat inventory icon
    Raw Meat
    Animal Hide inventory icon
    1 × Animal Hide

Sources