Ahmet Brkovic stuck just before the break and after City were denied by a post, Town secured their win with a Kevin Nicholls penalty in time added on.
After a subdued start by both sides Luton created the first threat when a Curtis Davies header from Dean Morgan's free-kick forced Rab Douglas into a diving save.
The next chance didn't arrive until the 21st minute when only a last-ditch intervention by Alan Maybury prevented Warren Feeney from having a clear shot on goal.
City had to wait until the 23rd minute for their first attempt on goal when Gareth Williams' long-range drive failed to find the target.
Leicester then went close when Momo Sylla's cross found Williams with only the keeper to beat from six yards. However, Marlon Beresford stood his ground to block the midfielders shot with his feet.
But it was Luton that broke through on 41 minutes with a wonder strike from Croatian midfielder Brkovic, who scored his third of the season.
Morgan's deep corner was headed back across goal by Feeney for Brkovic to execute a perfect overhead kick from seven yards which had Douglas grasping at thin air.
Williams had a great chance to level three minutes after the restart when his penetrative run saw him glide beyond Davies but his angled shot flew over Beresford but wide of the far post.
Dion Dublin then headed Sylla's cross narrowly wide as Leicester pushed hard for an equaliser. But it was the visitors that almost doubled their advantage on 65 minutes when Enoch Showunmi's knock-down found Paul Underwood but he blazed over from eight yards.
Williams almost unhinged Luton when his 12-yard strike appeared destined for the top corner until Beresford intervened to claw the ball away.
In the final minute City went agonisingly close to an equaliser when Dublin headed Jason Wilcox's cross against the foot of the post but there was no Leicester player on hand to turn home the rebound.
Then in the final minute of the four added on, Nicholls slotted home a penalty to secure the points for Luton after Maybury was ruled to have fouled Showunmi inside the box.