Jed Wallace had the chance put Millwall ahead from the penalty spot midway through the first half but fired over the bar, while Fulham nearly snatched a winner in the final seconds when Neeskens Kebano headed a corner onto the crossbar.
Scott Parker's side got off to the perfect start as Mitrovic slid Joe Bryan's inviting cross into the net.
The Serbian striker, the Championship's top scorer, has now netted in each of his three games against Millwall and it looked like it might be a long night for the home side against their in-form opponents.
But the Lions hit back within five minutes, with Bodvarsson firing past Marek Rodak after a corner.
The striker looked to be two yards offside but, after consulting, the officials determined the ball had reached him via the boot of Fulham's Josh Onomah rather than Tom Bradshaw's flick-on.
Millwall may have been lucky to be level but soon after they should have gone ahead.
Wallace tricked his way past two defenders before feeding Mahlon Romeo, who turned inside and was brought down inside the box. Andrew Madley pointed to the spot but Wallace hammered the penalty into the top tier.
Fulham looked rattled and barely mustered an attack for the rest of the half, their best effort a tame Mitrovic volley from long range that Bartosz Bialkowski gathered easily.
The promotion chasers came out far stronger after the break, with Onomah in particular troubling the home defence with his surging runs from midfield but twice failing to hit the target when well-placed.
But they rarely turned their dominance into chances and it was Millwall who looked like they might grab a winner when a rare counter-attack saw Bodvarsson set Bradshaw through on goal, only for the striker to shoot tamely at Rodak.
Fulham finally managed a clear-cut chance in stoppage-time when substitute Kebano met Ivan Cavaleiro's cross but his flicked header came back off the bar and was scrambled away.
The point draws Fulham back level with second-placed Leeds but out of the automatic promotion places on goal difference, while Millwall are now seven points off the play-off pace.