The Indian Ocean dipole is a prominent mode of coupled ocean-atmosphere variability, affecting the lives of millions of people in Indian Ocean rim countries. In its positive phase, sea surface temperatures are lower than normal off the Sumatra-Java coast, but higher in the western tropical Indian Ocean. During the extreme positive-IOD (pIOD) events of 1961, 1994 and 1997, the eastern cooling strengthened and extended westward along the equatorial Indian Ocean through strong reversal of both the mean westerly winds and the associated eastward-flowing upper ocean currents. This created anomalously dry conditions from the eastern to the central Indian Ocean along the Equator and atmospheric convergence farther west, leading to catastrophic floods in eastern tropical African countries but devastating droughts in eastern Indian Ocean rim countries. Despite these serious consequences, the response of pIOD events to greenhouse warming is unknown. Here, using an ensemble of climate models forced by a scenario of high greenhouse gas emissions (Representative Concentration Pathway 8.5), we project that the frequency of extreme pIOD events will increase by almost a factor of three, from one event every 17.3 years over the twentieth century to one event every 6.3 years over the twenty-first century. We find that a mean state change--with weakening of both equatorial westerly winds and eastward oceanic currents in association with a faster warming in the western than the eastern equatorial Indian Ocean--facilitates more frequent occurrences of wind and oceanic current reversal. This leads to more frequent extreme pIOD events, suggesting an increasing frequency of extreme climate and weather events in regions affected by the pIOD.