The Bahrain Victorious rider joined solo leader Igor Arrieta (UAE Team Emirates-XRG) at the front near the summit of the Montagna Grande di Viggiano climb, and when Arrieta took a wrong turn, he looked certain to take the win.
But Arrieta, banging his handlebars in anger, had other ideas and put down the power to reel in Eulalio along the finishing straight and win his first Grand Tour stage.
"I was completely empty after the hard stage but so was Eulalio," Arrieta said. "I kept pushing, and when I got his wheel, I knew he couldn't go faster than me and thought I could win the stage."
Eulalio had looked in control but slid to the tarmac with around 7 of the 203km remaining, allowing Arrieta to catch him. He then appeared to have been gifted the stage when Arrieta went the wrong side of a junction and had to turn around.
But Eulalio had nothing left in the legs on the final uphill drag and had to settle for second place, although taking the race lead was some consolation.
Arrieta had also crashed 13.5 km from the end of a cold and wet day that tested the resilience of the peloton.
Ciccone back down to sixth
Italian Giulio Ciccone, who had begun the 203-km stage in the pink jersey, finished seven minutes back in the main group and slipped down to sixth overall.
Eulalio now leads the GC standings by two minutes and 51 seconds from Arrieta. Christian Scaroni is in third place, 3:34 back.
Arrieta's victory was a welcome boost for his team, who lost GC contender Adam Yates, Australian Jay Vine and Spaniard Marc Soler, following a huge pile-up during the second stage in Bulgaria.
"I'm really, really happy to achieve this victory, it means a lot for me, because of the crash (on stage two) and all the team mates who went home," Arrieta said.
Thursday's sixth stage is a 142-km ride from Paestum to Napoli, passing near Mount Vesuvius.
