The Blues had tied it up 2-2 with 7:32 left in third period when Matt D'Agostini sent a pass from behind the net into the slot, where Vladimir Sobotka one-timed it past Jonathan Bernier.
But less than 2 minutes later, Mitchell's blast from the left point bounced off Blues center Scott Nichol and past a surprised Jaroslav Halak for the decisive tally.
Anze Kopitar and Mike Richards also scored for Los Angeles.
Bernier finished with 23 saves as the Kings won for the fifth time in seven games.
Alex Steen notched a first-period goal for St. Louis, which got 25 saves from Halak.
NBC reportedly holds celebs hostage to Jimmy Fallon's show
Ray Liotta sues skin care company over use of likeness