Showunmi, on as a 75th minute replacement for Peter Holmes, was ideally placed to lash a shot into the roof of the net from close range with just seven minutes remaining after Bournemouth had failed to clear a corner from the left.
The win - Luton's 11th on the road this season - moved the Hatters to within one win of clinching promotion and they are now well placed to take the League One title with second placed Hull due to play at Bradford on Sunday.
But the visitors were made to work hard for the three points and had to overcome the early loss of defender Russell Perrett to claim victory.
Referee Trevor Parkes had little option but to send the centre-back off for a professional foul after Perrett had tripped James Hayter in the 21st minute with the Bournemouth striker closing in on goal.
Perrett had performed wonders just 60 seconds earlier to clear a goal-bound shot off the line from Wade Elliottt after the Bournemouth winger had broken clear on the right and lifted a shot over goalkeeper Marlon Beresford.
Garreth O'Connor and Steve Fletcher wasted further good chances for the home side to take the lead before the interval, while Beresford leapt acrobatically to his left to save Stephen Purches' long-range shot from 30 yards on 38 minutes.
Bournemouth goalkeeper Neil Moss was in equally good form to deny Luton striker Steven Howard from 25 yards two minutes before the half-time whistle and the Hatters' top scorer continued to cause problems for the Bournemouth defence in the second half.
Howard headed over when well placed in the 66th minute while substitute Warren Feeney almost opened the scoring for Luton seven minutes later only to see his shot on the turn go narrowly wide.
Showunmi finally broke the deadlock much to the delight of Luton's travelling band of 1,300, fans but the visitors still required a fine save from Beresford to deny O'Connor to preserve three vital points.