On 2014 April 23, the Swift satellite detected a gamma-ray superflare from the nearby star system DG CVn. This system comprises a M-dwarf binary with extreme properties: it is very young and at least one of the components is a very rapid rotator. The gamma-ray superflare is one of only a handful detected by Swift in a decade. As part of our AMI-LA Rapid Response Mode, ALARRM, we automatically slewed to this target, were taking data at 15 GHz within six minutes of the burst, and detected a bright (~100 mJy) radio flare. This is the earliest detection of bright, prompt, radio emission from a high energy transient ever made with a radio telescope, and is possibly the most luminous incoherent radio flare ever observed from a red dwarf star. An additional bright radio flare, peaking at around 90 mJy, occurred around one day later, and there may have been further events between 0.1-1 days when we had no radio coverage. The source subsequently returned to a quiescent level of 2-3 mJy on a timescale of about 4 days. Although radio emission is known to be associated with active stars, this is the first detection of large radio flares associated with a gamma ray superflare, and demonstrates both feasibility and scientific importance of rapid response modes on radio telescopes.