The Serbian striker spent last season on loan with Fulham and last week signed a permanent deal with the newly-promoted side. New signings Fabricio, Jean Michael Seri, Andre Schürrle and Maxime Le Marchand also played a part for Fulham.
Slaviša Jokanović's side took the lead in the third minute through Mitrović, who beat Celta's offside trap, then continued pressing the Spanish defence.
Vigo equalised through Dennis Eckert's header in the seventh minute, though the pressure continued from Fulham and Aboubakar Kamara found the back of the net after a lay off from Mitrović to make it 2-1.
Celta pegged the 'Cottagers' back once more through a Gustavo Cabral header just before half time.
The tempo seemed to slow down late in the second-half with both sides failing to find a third goal. The match ended 2-2.