The Tasmanians built their lead to nearly 300 after the third day with Bevan going to stumps at 98 not out on Thursday.
The 34-year-old left-hander, who moved from New South Wales to accept an assistant coaching position after being dumped from the national one-day side, was the mainstay in a steady Tasmanian second innings which finished the day at 203 for four.
Tasmania claimed their first Shield points of the season, dismissing South Australia for 290 chasing 375, with left-arm spinner Xavier Doherty taking five wickets.
Doherty's victims included Graham Manou, after the South Australian captain led a spirited fightback with 97 from 105 balls.
In Hobart (day three): Tasmania 375 (M. Bevan 106, M. DiVenuto 91, D. Dawson 73; S. Tait 4-111, P. Rofe 3-50) and 203 for 4 (M. Bevan 98 not out, D. Marsh 22 not out; D. Cullen 2-77) vs South Australia 290 (G. Manou 97, D. Cullen 42; X. Doherty 5-66)