Brendan Rodgers' side fell behind to Clark Robertson's first-half header before Celtic defender Cameron Carter-Vickers scored an own goal just before the break.
Celtic fans had disrupted the game by throwing dozens of balls onto the pitch in their ongoing protest against the Parkhead board after a poor start to the season that included elimination from the Champions League qualifying stages.
Celtic's first league defeat this season was their first at Dens Park since Tommy Coyne scored the only goal of the game for the Dark Blues in 1988.
Booed off at the final whistle, Celtic remain five points behind surprise leaders Hearts, who they play at Tynecastle next weekend.
Before the trip to Edinburgh, Celtic have a Europa League tie against Sturm Graz on Thursday.
"It's not all linear and all smooth right the way through the season. That's for sure," Rodgers said.
"I think the challenge from the summer, now leading into here, where we lost a lot of firepower, a lot of goals out on the team.
"And there's no way you'll go into a race and be given the keys to a Honda Civic and say, 'I want you to drive it like a Ferrari'. It's not going to happen.
"So until something changes, I have to find the solutions. Because like I said, goals, speed, everything has come out of the team, and we need to find a way to be better.
"It was clear in the summer, but it's really the past now. There's nothing we can do about it."
The game came to a quick halt after the first whistle blew when Celtic fans bombarded the pitch with different coloured balls, causing a delay while they were cleared.
Celtic striker Kelechi Iheanacho clipped the outside of the post with a snapshot, but it was Dundee who took the lead in the 17th minute when unmarked Robertson rose to head home from a Cameron Congreve corner.
Kieran Tierney and Yang Hyun-Jun both missed the target with efforts as Celtic piled on the pressure.
However, in first-half stoppage-time, Congreve skipped past Celtic centre-back Liam Scales before crossing for Joe Westley to skim the ball in off helpless Carter-Vickers.
Celtic midfielder Reo Hatate fired wide from the edge of the area after the interval before Dundee substitute Ashley Hay went close with a header from 12 yards.
Dundee goalkeeper Jon McCracken twice thwarted Iheanacho, but Celtic's luck was out.
Hatate's shot appeared to be blocked by the arm of Paul Digby inside the Dundee area, and referee Matthew MacDermid immediately pointed to the spot.
But after he checked his pitchside monitor at the behest of VAR, he overturned his decision to leave Celtic frustrated.
