Scott Boden and Jacob Hazel scored for the Spireites who found themselves 3-0 down after 20 minutes.
Liam Cooper gave Hull a fifth minute lead with a firm near-post header from Jamie Devitt's corner and Jonathan Margetts (17) made it 2-0 following a searching through ball from Conor Townsend.
Lee made a couple of fine saves before Townsend (20) put the visitors further ahead with a close-range finish.
Chesterfield then enjoyed a good spell with Jonathan Wafula putting a couple of free headers off-target before Boden (35) pulled a goal back with a fine turn and shot from the edge of the area.
Early in the second-half, Tim Whittaker's long-range effort just cleared the bar while Cameron Mason, on as a replacement for Lee, kept out a low shot from Sam Topliss.
Hull regained their three-goal lead when a corner was headed back at the far post and Margetts (67) found the net with a volley.
Hazel scored a minute later, producing a neat finish after being sent clear by Mark Randall.
Towards the end, Boden put a 20-yard effort over and Mason made a good save to deny Lewis Clarkson.
Coach John Dungworth said: "We got off to a terrible start. We were all over the place defensively and we were fortunate to keep it down to three because Tommy Lee made some outstanding saves. Tommy made an excellent return. He looked very sharp, composed and confident and he's obviously in the manager's plans for the weekend.
"I thought the lads fought back well. Scott Boden took his goal really well and Jacob produced a good finish. Micah Evans did really well down the right-hand side. Jonathan Wafula had two free headers and should have done better."
The Chesterfield line-up featured triallist Oscar Radford in defence. Radford is a former Sheffield United trainee who also had a spell at Doncaster Rovers before linking up with Kettering. Dungworth was impressed with his performance, commenting: "Oscar did well. He trained yesterday, settled in well and the manager has invited him to train on Thursday."
Chesterfield line-up: Lee (Mason 53), Broadhead, Whittaker, Radford, Trotman (Scully 45), Randall, Evans, Wafula, Westcarr (Hazel 45), Boden, Waddle.