Her parents were Hungarian immigrants who met when they got to Adelaide, and married after her father got a job as an Opal miner. Jancso has an older brother. The family spent many years in Coober Pedy.
Changing her actors union name to Annie Jones, she undertook some modelling, and began acting aged 17 in the role of Chrissie in the drama Run Chrissie Run!. She then appeared in The Henderson Kids (1985), and Sons and Daughters (1986) as Jesse Campbell.
Having auditioned for the role of tomboy Charlene Mitchell, which she lost out to Kylie Minogue; aged 19, she landed her role in Neighbours (1986–89) which brought her a high public profile. This culminated in the Logie Award for Most Popular Actress in 1989. Jones played the title role in the drama Jackaroo (1990), which won her another Logie award – 1991 Most Popular Actress in a Telemovie or Mini-series. She also starred as Eva Kovac in the historical mini-series Snowy (1993).
In comedy, she co-starred in the Australian sitcom Newlyweds (1993–94) and guest starred in The Adventures of Lano & Woodley (1997). In children's TV, she played the mother in Pig's Breakfast (1999) and appeared in the Mortified episode Mother in the Nude (2006). Annie has also guest starred in many Australian series including City Homicide (2007), Stingers (2003), Blue Heelers (2002), Marshall Law (2002), Good Guys, Bad Guys (1997) and Halifax f.p.(2006).
In 2010 she was cast in the first of three new The Underbelly Files television movies. The first two-hour telemovie, Tell Them Lucifer Was Here, tells of the biggest criminal investigation in the history of The State of Victoria.