Yates picked up three more seconds on defending champion Tom Dumoulin by taking bonuses from the intermediate sprint, but saw Mitchelton-Scott team-mate Esteban Chaves lose second place overall and drop out of the top 10 in a blow for his team.
Froome came into the season with a double objective of winning the Giro d'Italia and Tour de France, something that has not been done since 1998.
"We decided before the stage that we wouldn't chase for the stage, because normally in a finish like that there are some faster guys who have won before me", Yates said afterwards.
"I'm not going to lie, that crash took a lot out of me". But with two sprint stages ahead before a mammoth stage 14 on Saturday night, a solid grip on sixth place is an extremely advantageous spot to be in as the race starts to get serious. "If I lose more time on the Zoncolan or the day afterwards, then it would be very hard", Dumoulin said.
Thursday morning's headlines will belong inevitably to Yates, who accelerated with notable facility on the 16% gradient of the Via Olimpia, and barely relented from there to the finish. "But I have to keep trying because I still don't have enough". "I'm going to keep fighting".
Mohoric thanked his team, which is centered around Domenico Pozzovivo in fourth overall, for letting him go for the victory.
"I certainly haven't given up hope, " Froome said.
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The hilly stage included a climb through Michele Scarponi's hometown of Filottrano in honour of the 2012 Giro victor who died after a collision with a van while training shortly before last year's race. Three minutes from the leader of the General standings are only eight riders.
There were a number of early attacks but nothing stuck until Luis León Sánchez and Alessandro De Marchi took off.
Mirco Maestri and Alex Turrin made it over to the leaders with about 100km remaining and the gap hovered around 3 minutes.
Zdenek Stybar and Tim Wellens attacked and opened up a small gap late in the race but couldn't respond when Yates made his move.
The stage was raced on a hilly route over 156 kilometres from the central city of Assisi across to Osimo near the Adriatic coast.
The race ends in Rome on May 27.