Cook crashed home a spectacular volley 14 minutes before half-time to set the Cherries on course for a first league win over their south coast rivals in 58 years.
And Benik Afobe, a £10m signing from Wolves in the transfer window, sealed the win when he headed in a Matt Ritchie free-kick at the near post after 79 minutes.
Austin – who rose to fame playing for Bournemouth’s non-league neighbours Poole Town – had earlier missed a sitter before hobbling off injured.
A competitive derby clash was held up by fourth official Kevin Friend being KO’d after hitting his head on the Southampton dugout.
The bizarre incident forced referee Mike Dean to call an early halt to the first half with Friend taken to hospital as a precaution.