It's been a topsy-turvy week for Luton, who netted five themselves in beating Leeds eight days ago, but then slumped 4-0 at Everton in the Carling Cup in midweek.
The visitors have been suffering from a flu virus and injuries, but even at full-strength they would have struggled to contain Ipswich, who recorded their biggest win under rookie boss Jim Magilton.
Sylvain Legwinski glanced a header home in the first half to put Ipswich ahead and they could have been out of sight by half-time, such was their complete control of the action.
Rowan Vine controversially had a goal chalked off for offside, before Lee headed down for Jaime Peters to put Town 2-0 up, again with a hint of controversy, but after that there was only one winner.
Town recorded over 20 attempts at goal and as well as the five they netted they also hit the woodwork twice, while Marlon Beresford, who had little chance with the goals, made some fine stops.
Legwinski and Steve Robinson for Luton shot well over in a messy start to the game, before Billy Clarke blazed a good chance over after Jason De Vos headed down a Matt Richards corner.
A silky passing move saw Richards and Peters involved before Harding raced into the box, only to drag his left-foot shot wide.
Town were unlucky on 19 minutes when a Richards corner was only half cleared and Peters crossed for an unmarked Alex Bruce to thump a header against the angle of post and bar.
However, Town weren't down for long as Richards swung in another dangerous corner moments later and Legwinski's near-post glance beat the efforts of Keith Keane to clear at the far post.
In a brief Luton rally, Lewis Emanuel shot over before Warren Feeney fed in a dangerous cross just out of the reach of Carlos Edwards. But it was soon all Ipswich again.
Beresford produced a fine diving save to turn aside a Simon Walton curler, but he was beaten on 32 minutes when Peters and Clarke combined to find Legwinski and his low 22-yarder thudded off the base of the upright.
A Walton volley deflected narrowly over off Keane, Beresford dropped onto a Legwinski shot on the line and, just before the interval, Clarke raced clear onto a Lee flick only to see Beresford knock down his shot, before the teenager blazed the rebound over.
Luton should have been level on 47 minutes when Edwards escaped Dan Harding down the right flank and crossed for an unmarked Vine to score but he was wrongly flagged offside - the final ball being crossed when Vine was behind his team-mate.
That proved crucial on 54 minutes when Beresford, who had just saved well again from Clarke, spilled a Harding cross under pressure from Lee and Peters was in the right place to tap in his first Ipswich goal. Some referees may have ruled it out as Lee's arm was across the goalkeeper.
Lee netted his first on 66 minutes when Legwinski won a loose ball and poked it through for the Republic of Ireland international to outpace Kevin Foley and slide home an angled shot.
Luton finally had a legitimate shot on target on 67 minutes, when Shane Supple tipped away a Robinson corner, then saved a Vine header which was straight at him.
Ipswich had the ball in the net again on 72 minutes, but the flag went up as sub Gary Roberts pulled the ball back for fellow replacement Danny Haynes to score, the ball having gone out of play in the build-up.
Haynes had a hand in the next goal though, Beresford pushing away his header from a Richards corner straight to Lee to fire in.
Lee sealed his treble from the penalty spot in the final minute after Foley brought down Roberts, albeit after the winger had handed Foley off in the face.