The Dockers moved to fourth on the log on 30 points following the win.
Felly Mulumba gave Bandari the lead in the ninth minute from the spot after William Wadri was brought down inside Chemelil Sugar box.
Goalkeeper Farouk Shikhalo denied Chemelil a chance to level the scores after the former Posta Rangers custodian saved a penalty just a few minutes later to keep his 11th clean sheet this season.
Bandari’s victory stretched Chemelil winless run to six games, leaving the Millers just two places above the relegation zone.
Bandari took their run to two straight victories having beaten Vihiga United in their last game before Tuesday’s triumph.