Emmanuel-Thomas fired home with the aid of a deflection after 35 minutes after Brendan Moloney had broken clear on the right and pulled the ball back to the edge of the box.
City missed a catalogue of chances to sew up the result before substitute Marvin Elliott forced home a Marvin Pack cross in the 87th minute.
Emmanuel-Thomas set the seal on an outstanding personal contribution by lashing home the third from the corner of the penalty area in time added on.
Dagenham had created the better chances early in the game. Luke Howell had a 19th-minute shot cleared off the line by Aden Flint and forced a diving save from City keeper Elliot Parish with a low shot.
With Emmanuel-Thomas having an increasing influence, City began to pass the ball more incisively. Scott Wagstaff had a 43rd-minute shot blocked and Nicky Shorey wasted a good opportunity by dwelling on the ball.
The second half saw the home side dominate, getting behind the Daggers defence on countless occasions.