The 59-year-old will head a panel of three executive vice-chairmen which features Baker, Alex Rowe and ex-managing director Bill Phillips.
Bristow won £15million on the National Lottery with her late husband Paul in 2004 before funding the takeover of the Gulls three years later.
She told the Western Morning News: "My midfield of vice-chairs will keep me in line, but I will hopefully catch on quick and not score too many own goals."
Former chairman Baker added: "She is a very bright woman who, because she's not a businesswoman as such, sees things from a much better angle than the rest of us do sometimes.
"This hasn't been a sudden decision. Thea is the logical chair simply because of her tremendous financial support for the club."