After Anderson struck in the 17th minute, Bauza completed a 19-minute treble in the first half and though the visitors replied through Wayne Gray, the Swans fully deserved their victory.
There were six changes to the Swansea side beaten by Yeovil, including a return for skipper Gary Monk.
A mistake by Charlie Daniels gave Anderson a free run into the danger area and he began the scoring with a low shot into the corner of the net.
The Swans went further ahead in the 25th minute when a cross from Anderson was held up by Jason Scotland despite being surrounded by two Orient defenders. He held on until Bauza raced in to slide the ball home.
The Spaniard claimed his second ten minutes later when the visitors defence stood still waiting for an offside flag that never arrived.
Orient keeper Glenn Morris raced out of goal to clear from Scotland, but the ball ran to Bauza who slotted into an empty net from fully 25 yards.
His hat-trick came in fortunate circumstances in the 44th minute when Andy Robinson's shot was pushed out by Morris. Bauza messed up but his shot struck the crossbar and bounced down on to the line and the referee's assistant signalled a goal.
Scotland should have added to the score seven minutes after the break, but Morris went down bravely at the feet of the striker to push the ball behind for a corner.
Gray pulled a goal back for Orient when he rose above Angel Rangel to head in a JJ Melligan cross in the 63rd minute.
Both sides could have added to their score in an entertaining game, but the Swans fully deserved their victory.