Hull took a first-half lead when Sam Ricketts swung in a cross which Livermore powerfully headed in.
The visitors doubled their advantage in spectacular fashion just after the hour mark as Livermore crossed and Turner smashed home a superb volley.
Luton hit back nine minutes later when substitute Dean Morgan's free-kick was headed in by Drew Talbot.
Luton manager Mike Newell on how the sale of Rowan Vine, Kevin Nicholls, Carlos Edwards and Steve Howard has disrupted his team:
"Give that to me and I'll get five experienced players in, I'll pay them fortunes and we'll get ourselves out of it.
"It takes three or four years to build a reputation as a decent manager and three months to lose that reputation. But if you take the four best players out of any team it will hurt them.
"Give me the money and I'll replace them - but I can't. The reason we're in this position is because we've sold our best players."
"We came with a game plan and played to it until we went two up and then it went out of the window.
"So the lesson is to stick to the game plan for 90 minutes, and now for the last eight games."
Luton: Beresford (Brill 33), Foley, Barnett, Heikkinen, Davis, Bell, Keane, Langley (Feeney 46), Emanuel (Morgan 56), Carlisle, Talbot.
Subs Not Used: Coyne, Runstrom.
Booked: Carlisle, Barnett.
Goals: Talbot 70.
Hull: Myhill, Ricketts, Turner, Delaney, Dawson, Parlour (Coles 88), Ashbee, Livermore, Elliott, Windass (Barmby 80), Forster (Vaz Te 19).
Subs Not Used: Duke, Marney.
Booked: Dawson, Livermore, Vaz Te, Elliott.
Goals: Livermore 22, Turner 62.
Att: 7,777.
Ref: R Olivier (W Midlands).