The Porto winger struck in the 66th and 94th minutes to take his personal tally for the tournament to four goals, leaving him as the joint top scorer alongside Lionel Messi on Friday.
"We've been able to come together as a team and show great unity all throughout the tournament," Diaz said after the match. "It makes me very proud to be a part of this team and achieve what we have."
Yoshimar Yotun opened the scoring for Peru just before halftime when he received Christian Cueva's slide-rule pass and slotted a low shot past goalkeeper Camilo Vargas.
Juan Cuadrado equalised just after the restart, bending a low free-kick from the edge of the box around the wall and inside the near post.
Diaz continued his impressive run of form by latching onto Vargas's 60-yard goal-kick before a surging run and low drive into the bottom right corner from just inside the 18-yard box.
Peru did not relent and they were back on level terms when Gianluca Lapadula headed home from a Raziel Garcia corner, eight minutes from time.
But Colombia were jolted into action again as Peru were pushed deep into their own half in the final minutes.
Diaz prompted wild celebrations in the fourth minute of stoppage time with his second bid for goal of the tournament in as many games, lashing a wicked long-range effort into the top right corner after Luis Muriel released him with a backheel.