Xojiakbar Alijonov scored the decisive header in the 120th minute at the Olympic City Stadium in Tashkent. Eldor Shomurodov had missed a penalty in the 105th minute.
Iran went down to 10 men early in the match when Arya Yousefi was shown a red card just five minutes in.
Earlier in the day, India defeated Oman 3-2 in a penalty shootout after the match ended 1-1 in regular time.
Iran began the campaign in Group B with a 3-1 win over Afghanistan and then defeated India 3-0. Team Melli and Tajikistan drew 2-2, but Iran topped the group to advance to the final.
Iran had won the inaugural CAFA Nations Cup two years ago, edging Uzbekistan 1-0 in the final.
FIFA President Gianni Infantino attended the match.