This was the toughest stage in the Giro so far and it marked the collapse of top favorite, Marco Pantani.
Pantani, reputedly the best Giro climber, ended nearly seven minutes behind the two top finishers at one of the highest passes in the Dolomite mountains.
Denied a spot in the upcoming Tour of France, Pantani had been hoping of an outstanding performance in the Giro to have the French organizers changing their mind, giving him a wild card entry.
Simoni snatched the pink jersey from fellow Italian Dario Frigo who trailed Cuapio and Simoni by 45 seconds, in third place.
Colombian Hernan Buenahora, of Selle Italia, placed fourth, ahead of compatriot Carlos Contreras Cano and Spain's Unal Osa Eizaguirre.
Frigo, of Fassa Bortolo, had entered Friday's stage with a thin one-second lead over Simoni, a 30-year-old who wore the pink jersey for the first time in his career.
"It's a dream coming true,'' Simoni said at the finish line of the stage named after late Italian "Campionissimo'' Fausto Coppi.
"I will have to work hard to retain the pink jersey. However I'm ready to attack again from the next stage. I believe that Frigo will be my toughest opponent for the title,'' the race leader added.
Pantani, a Giro and Tour of France champion in 1998, slipped nearly three minutes behind as the road climbed to the Fedaia pass, the third and next-to-last pass of the stage.
The Mercatone Uno captain recovered nearly one minute down the descent but lost more ground in the final 12K (7.4-mile) climb to the Pordoi Pass. He placed 32nd, 6:46 minutes behind Perez Cuapio.
"It's a day to forget, a bad day for me'' Pantani said. "My legs were not working. There was no a particular reason but I was unable to keep the the pace of the others.''
Pantani, dogged by doping accusations since 1999, definitely spoiled his chances for a Giro victory as Simoni and Frigo are both good climbers and strong enough in time trials.
A 55.5K (34.4-mile) time trial is scheduled on Sunday, following another mountainous stage Saturday.
The 21-leg Italian cycling marathon ends in Milan on June 10.
The big losers of the day included defending Giro champion Stefano Garzelli who finished 13:52 minutes behind the winner.
Simoni and Perez Cuapio broke away from Frigo's small group 10K from the finish.
Simoni, certain of the race leadership, let the Mexican cyclist sprint to victory in the last 50 meters and contented himself with the second place.
Perez Cuapio, of the Ceramiche Panaria team, completed the grueling 225K (139.5-mile) leg from Montebelluna to the 2,239-meter high Pordoi Pass in seven hours, 24 minutes, 48 seconds.
Before Friday's win, the Mexican cyclist had been victim of a streak of misfortunes.
Perez Cuapio broke the chain while pedaling to victory in the final kilometers of the fourth stage and knocked out his two front teeth following a bad spill during the seventh leg.
In the overall standings, Simoni led Frigo by 48 seconds and third-place Wladimir Belli by 1:07 minutes.
Spain's Abraham Olano, who trailed the stage winner by 3:02 minutes, slipped from third to eight overall, with a gap of 3:23 minutes.
Buenahora moved to sixth overall, 2:19 behind Simoni.
Saturday's 14th stage, a 163-kilometer (101.06-mile) ride from Cavalese to Arco, in northereast Italy, includes the Monte Bondone and Santa Barbara climbs.
Stage 13: 225K from Montebelluna to Passo Pordoi:
1. Julio Perez Cuapio (Mex) Panaria 7hr 24min 48sec
2. Gilberto Simoni (Ita) same time
3. Dario Frigo (Ita) at 45
4. Hernan Buenahora (Col) 48
5. Carlos Contreras (Col) 49
6. Unai Osa (Spa) 57
7. Wladimir Belli (Ita) 1min 01sec
8. Ivan Gotti (Ita) 1:03
9. Joaquim Castelblanco (Col) 2:16
10. Abraham Olano (Spa) 3:02
11. Andrea Noe (Ita) 3:02
12. Serhiy Honchar (Ukr) 3:02
13. Marzio Bruseghin (Ita) 4:15
14. Peter Luttenberger (Aut) 4:46
15. Paolo Savoldelli (Ita) 4:46
16. Pietro Caucchioli (Ita) 4:46
17. Tadej Valjavec (Slo) 4:46
18. Aleksandr Shefer (Kaz) 4:46
19. Pascal Herve (Fra) 4: 46
20. Jose Azevedo (Por) 4:46
30. Danilo Di Luca (Ita) 6:25
32. Marco Pantani (Ita) 6:46
47. Stefano Garzelli (Ita) 13:52
108. Jan Ullrich (Ger) 30:31
1. Gilberto Simoni (Ita) 63hr 23min 15sec
2. Dario Frigo (Ita) at 48sec
3. Wladimir Belli (Ita) 1min 27sec
4. Unai Osa (Spa) 1:52
5. Ivan Gotti (Ita) 2:14
6. Hernan Buenahora (Col) 2:19
7. Carlos Contreras (Col) 2:36
8. Abraham Olano (Spa) 3:23
9. Andrea Noe (Ita) 3:53
10. Jose Azevedo (Por) 4:56
11. Serhiy Honchar (Ukr) 5:03
12. Pietro Caucchioli (Ita) 6:01
13. Giuliano Figueras (Ita) 6:38
14. Aleksandr Shefer (Kaz) 6:47
15. Joaquim Castelblanco (Col) 6:49
16. Peter Luttenberger (Aut) 7:21
17. Danilo Di Luca (Ita) 7:29
18. Marco Pantani (Ita) 8:04
19. Marco Magnani (Ita) 8: 05
20. Marco Velo (Ita) 8:10
21. Paolo Savoldelli (Ita) 8:16
23. Pascal Herve (Fra) 9:16
32. Stefano Garzelli (Ita) 14:58
80. Jan Ullrich (Ger) 54:42
119. Mario Cipollini (Ita) 1hr 17min 48sec