The South Korea international capped an impressive personal performance by curling home a spectacular 20-yeard free-kick on 36 minutes.
The opening exchanges provided little in terms of goalmouth action, with both sides struggling to carve out openings.
Roman Bednar and David Worrall came closet to breaking the deadlock for the Premier League side, before Kim showed them the way to goal.
The inevitable clutch of substitutions at the break ensured that the game once again lacked fluidity at the start of the second half, but it was the hosts who settled first.
Liam Davis saw an ambitious lob sail narrowly over the bar, while Colin Larkin saw a strong header pushed away by Dean Kiely.
Leon Constantine also saw a low drive deflected away by a desperate block from Tininho as the Cobblers continued to push for an equaliser.
Tony Mowbray's men did finish the game the stronger, though, with former Dutch Under-21 international Sherjill MacDonald looking lively after stepping off the bench.
However, there were to be no more goals and West Brom secured a confidence boosting victory ahead of their return to top flight action.