Mickey Mellon's side arrived at Kenilworth Road on Thursday knowing a two-goal victory would see them replace Wycombe Wanderers as league leaders.
However, Shrewsbury could consider themselves somewhat fortuitous to come away with a point as Luton enjoyed the better of proceedings.
The hosts hit the post through Jake Howells early on and applied significant late pressure following the 70th-minute dismissal of Shrewsbury defender Nathaniel Knight-Percival for a second bookable offence.
Yet visiting goalkeeper Jayson Leutwiler held firm, making a number of impressive stops to ensure the match finished goalless.
Shrewsbury remain third in the table, behind Wycombe and Burton Albion having played a game more, while Luton leapfrog Newport County into fourth.
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