Andrews, who stepped up from set-piece coach to replace Thomas Frank in June, has impressed with victories over Manchester United and Liverpool this season. The 45-year-old Irishman said he wanted to "earn my stripes" after following Frank's successful tenure, with the Bees now eyeing European qualification for the first time in their history.
Club director Phil Giles confirmed the extension was reviewed within nine months, describing Andrews as a "really good fit" after an "outstanding" first season. Brentford return to Premier League action on Saturday when they host Burnley.