A member of the day's breakaway, Katusha's Zakarin went solo in the final climb to beat Spain's Mikel Nieve by 35 seconds and another Spaniard, Mikel Landa, by 1:20 with Jan Polanc of Slovenia retaining the overall leader's pink jersey.
Slovenian Roglic and Italian Nibali crossed the line 2:57 off the pace after allowing several riders to come back into the game.
"The last climb was really hard, I did not have a lot of strength left," said Zakarin, who now has ambitions for the general classification.
"Now I can aim at a top five."
It was another bad day for Simon Yates as the Briton finished five minutes behind Zakarin after he had already lost considerable ground in last Sunday's individual time trial.
Mitchelton Scott's Yates cracked after Landa attacked early in the last climb, a 20.3-climb up to Lake Serru, the Movistar rider benefiting from the help of several team mates.
Another top contender, Colombian Miguel Angel Lopez, dropped out in the last ascent after suffering mechanical problems and reached the finish 4:19 down on Zakarin.
Overall, Polanc leads Roglic by 2:25 with Zakarin third 2:56 off the pace.
Dutchman Bauke Mollema is fourth, 3:06 behind Polanc, with double champion Nibali 4:09 down on the leader.
Saturday's 14th stage is a gruelling 131-km mountain trek from Saint Vincent to Courmayeur featuring five categorised climbs.
(Reporting by Julien Pretot; Editing by Pritha Sarkar)
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