Jagielka opened the scoring after 34 minutes, turning in after Paul Ifill's shot had been blocked.
Chris Morgan doubled the lead four minutes after the break, heading home Alan Quinn's free-kick.
Sub Vincent Pericard scored a tap-in after 71 minutes and Jagielka seized on a loose ball to add a fourth from 15 yards with only two minutes left.
Luton manager Mike Newell:
"I don't think 4-0 does it justice but at the end of the day it's 4-0, so we have to take it on the chin.
"I'm not that unhappy to be quite honest. If we had come here and been given a good hiding and not had any of the play, then I might be worried.
"But it was against a side that is leading the way in the division and scoring goals. But I'm not unduly worried with the performance and I thought for the first 20-25 minutes we were well in the game.
"They are the best side we've played and the most effective, so I've got no complaints. We've been beaten by a more effective side."
Sheff Utd: Kenny, Bromby, Morgan, Jagielka, Unsworth, Ifill, Alan Quinn, Montgomery, Geary (Wright 74), Shipperley (Pericard 61), Kabba (Webber 76).
Subs Not Used: Tonge, Gillespie.
Goals: Jagielka 34, Morgan 50, Pericard 71, Jagielka 88.
Luton: Beresford, Foley, Coyne, Underwood (Keane 73), Heikkinen, Edwards, Brkovic (Showunmi 66), Holmes, Barnett, Feeney (Vine 66), Howard.
Subs Not Used: Morgan, Brill.
Ref: M Dean (Cheshire).