Greg Halford's early goal secured the three points for Forest at The Den and Republic of Ireland goalkeeper David Forde also denied Radoslaw Majewski and Marcus Tudgay, after the interval.
But the Lions had their chances, too, with winger James Henry blazing over and hitting the side-netting before Karleigh Osborne headed a late chance narrowly wide.
Millwall could also have had a penalty when Shaun Batt raced through and went down under Elliott Ward's challenge, but referee Kevin Wright ignored the appeals.
The south Londoners have won just one of their last seven matches in what has been a tough end to the season following the disappointment, on and off the pitch, of their FA Cup semi-final defeat by Wigan.
Jackett said: "It was a poor goal to give away from a set-piece.
"We had plenty of time to get back into it and I can't fault the commitment of the players but we needed a break, something to fall for us.
"Ultimately it's disappointing as you need to score goals at home and creatively we weren't quite there today.
"We've been close but not close enough, which has been the story of the second half of our season."