Victoria decimated Tasmania's top order to leave the home team flailing at 46 for 6 in the hour before tea on Wednesday.
But Bevan (95) and Wright (99) joined forces to post an unbroken 191-run stand by stumps, leaving Tasmania at 237 for six -- 257 shy of the massive 494 required for victory.
Michael Lewis claimed 4-18 off 7.4 overs.
Victoria were keen to claim their first-ever victory against Tasmania in Hobart after already securing first-innings points.
In Adelaide, a fine double by all-rounder Shane Watson helped give Queensland almost total control of their match against South Australia at Adelaide Oval.
Watson (4-25) and Andy Bichel (4-71) routed SA for 150 before the Australian one-day allrounder scored 61 as he and Jimmy Maher (80) put on 146 for the second wicket at better than a run-a-minute.
Queensland reached 237 for four in their second innings, an overall lead of 366 with two days remaining.
By stumps Andrew Symonds had belted a six and eight fours in a typically belligerent unbeaten 51 off 53 balls.
Watson's allround contribution was a welcome sight for Queensland and national selectors, after a stress fracture in his back restricted him to just 58 overs in his 10 Shield matches last summer.
In Hobart (day three) : Victoria 162 (I. Harvey 79, M. Inness 24; A. Griffith 7-54) and 432 for 9 declared (G. Rummans 188, B. Hodge 96, D. Hussey 67; A. Downton 5-94, A. Griffith 3-119) v Tasmania 101 (M. Bevan 19, J. Cox 16, M. Di Venuto 16; M. Lewis 3-13, I. Harvey 3-33) and 237 for 6 (D. Wright 99 not out, M. Bevan 95 not out; M. Lewis 4-46).
In Adelaide (day two): Queensland 279 (J. Hopes 57, A. Noffke 44 not out, A. Symonds 44; P. Rofe 5-93) and 237 for 4 (J. Maher 80, S. Watson 61, A. Symonds 51 not out; S. Tait 3-73) v South Australia 150 (G. Blewett 37, T. Plant 31; S. Watson 4-25, A. Bichel 4-71).