Both clubs started the season with a League-imposed deficit of 17 points for going into financial administration last season.
Rotherham have quickly got rid of their handicap, but after this result Bournemouth are still seven points adrift.
Bournemouth were the dominant side, particularly in the first half when they gave Rotherham little scope to attack.
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Rotherham goalkeeper Andy Warrington flew to his right to deny Anderton in the 23rd and Rankine had a tame shot cleared from the line in the 32nd by Andy Nicholas.
Warrington saved Rotherham again in the 43rd minute when he blocked a shot from Joel Ward with his legs, before Rotherham at last sprang to life.
In first-half injury time a shot through a crowd of players, Rotherham's first effort on target, was smothered at the last second by home goalkeeper Shwan Jalal.
The second half was more even and Rotherham almost snatched a surprise lead in the 58th minute when Jalal did well to block a shot, again through a crowd of players from substitute Marc Joseph.
Anderton should have put Bournemouth ahead in the 63rd minute when Lee Bradbury crossed from the left, but from 12 yards Anderton somehow shot over the bar as he ran onto the pass.
Anderton's other potential match-winning moment came in the 87th minute, but from the edge of the area his low right-foot shot hit the inside of a post before rebounding to safety, with goalkeeper Warrington a helpless bystander.