Both Heckingbottom and Lions boss Gary Rowett felt Wednesday night’s exertions in the wins over Blackburn Rovers and Derby County respectively, had caught up with their sides at the Den.
Heckingbottom said: “It was an attritional game [between] two teams who played Wednesday night and I thought first half that’s what it looked like.
“From our point of view, I don’t think there was anything in it.
“If you look at statistics and passes, and all them things that don’t really matter, we probably shaded it.
“We’d have taken a 0-0 the way we played and come away with a point from here, with the form that Millwall are in.
“But, performance-wise, we didn’t think we were at our best and that’s the disappointing thing: the way we lost.
“I thought we could have been better, first half – slicker, sharper quicker.
“When it was there for us to go and create, we didn’t and then we were moving the ball quicker, that’s when they had the lead and were defending very well.”
After a goalless first half, which was a tough watch, the winning goal came in the 61st minute when Jed Wallace’s pinpoint free-kick from the right was headed in at the back post by the unmarked Jake Cooper.
It meant a fourth win in a row for the Lions and manager Gary Rowett said: “It’s quite phenomenal, really, to play a team like Sheffield United, after Wednesday night’s exertions [at Derby].”
Millwall: Bialkowski, Hutchinson (Ballard 64), Cooper, Murray Wallace, McNamara, Mitchell, Saville, Malone, Jed Wallace, Burey (Lovelace 90), Bennett (Evans 16). Unusued substitutes: Long, Pearce, Mahoney, Afobe.
Sheffield United: Foderingham, Gordon (Berge 71), Egan, Robinson, Baldock, Norwood, Hourihane (Fleck 87), Norrington-Davies, Ndiaye (Gibbs-White 56), McBurnie, Sharp. Unusued substitutes: Adam Davies, Ben Davies, Jebbison, Starbuck.
Referee: M Salisbury (Lancashire).