The match was played in sweltering heat at Solomon Mahlangu Stadium and, when extra time could not separate the teams, spot-kicks were needed.
The 2019 winners, Galaxy, will play Durban City in the final on May 2 after the latter edged second-tier Casric Stars 1-0 in their semi-final on Saturday.
Milford took the lead in the 11th minute when Bulelani Sigomoshe broke down the right and his low cross was turned into the net by Siphosethu Ndlabi with a clever backheel finish.
It was a second Nedbank Cup goal of the season for Ndlabi, who also netted in the Last 16 victory over Mkhambathi.
It was also the first goal that The Rockets have conceded in this year’s competition after three successive clean sheets.
The goal came against the run of play after Galaxy had made a bright start to the game, and the home side were level on 55 minutes with veteran Victor Letsoalo finding the back of the net.
Letsoalo headed into the net from Seluleko Mahlambi’s cross from the left.
He took his career tally in the competition to nine, having also netted against Marumo Gallants earlier in the season.
In the first semi-final on Saturdayl, City profited from an early own goal by Sphamandla Mlilo to book their second Nedbank Cup decider after losing to Free State Stars in the 2018 final.
Galaxy were champions seven years ago when, as a second-tier side, they defeated Kaizer Chiefs in the final in Durban. They will be hoping for a second title, while City are seeking a first victory in South Africa’s FA Cup.