Tasmania went into the four-day game with three of their most experienced batsmen -- Jamie Cox, Michael Di Venuto and Dan Marsh -- out through injuries but ended the day on 266 for seven.
Acting captain Damien Wright was on 38 and Xavier Doherty was on 13 at stumps.
Bailey, in his first-class debut, made 70 off 105 balls with 11 fours after Tasmania had slipped to 122 for five just before tea.
Opener Dawson, in his fifth first-class match, made 65.
For NSW, former Test leg-spinner Stuart MacGill took 3-83, left-arm medium pacer Nathan Bracken 2-30 and off-spinner Jason Krejza 2-63.
Michael Bevan's return to his old home ground turned into an anti-climax when he was out for 21, caught by MacGill off Krejza.