The American, the champion in Houston and Newport, has been in excellent form this week and broke the serve of second seed Carreno Busta three times in a 6-3 6-4 victory.
In the final, Johnson will take on Russia’s Daniil Medvedev, who eased to a 6-1 6-1 victory over Japan’s Taro Daniel.
Dimitrov through after Carreno Busta falls ill
Medvedev, 22, won his first ATP Tour title earlier this year in Sydney.
Nadal survives scare to reach Paris semi-finals
Djokovic battles back to beat Carreno Busta and reach semi-finals