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The visitors almost opened the scoring in the early minutes of the game when Junior Tallo unleashed a shot from outside the box, only to see his effort smash back off the crossbar.
Newly-promoted Metz were sparked into life by the chance and they took the lead in the 14th minute when Krivets was found by Kevin Lejeune, leaving the midfielder to fire home into the top corner.
Tallo came close to restoring Bastia parity but could not keep his headed effort down, while Yannick Cahuzac dragged a speculative effort wide of the post.
Despite their wasted chances, Bastia did claw their way back into the game just before half time when Cahuzac measured a pass to Floyd Ayite, who tucked in from the centre of the box.
However, Metz retook the lead 20 minutes into the second half as Milan converted into the roof of the net following a corner.
The hosts made sure there was no coming back for Bastia when Falcon made it 3-1 with 15 minutes to play, with the striker finishing neatly following a fast break from Krivets.