Union took the lead late in the first half thanks to a Josip Juranovic free-kick, but Bochum's Kevin Stoeger equalised from the spot early in the second.
Union were reduced to 10 men shortly after the penalty but were the more dangerous side in the closing stages, forward Kevin Behrens going agonisingly close to a winner with five minutes remaining.
Despite dropping points, Union leapfrogged RB Leipzig who won 3-2 at home against Augsburg on Saturday to move back to third. Free-kick specialists Union opened the scoring in typical fashion in first-half injury time.
Croatia international Juranovic hit a perfectly weighted free kick between a gap in the wall and past 'keeper Manu Riemann, his fourth goal since arriving from Celtic in January.