Mark Robins' men were high on confidence after victory over Morecambe last week and it showed early on with Marc Newsham hitting the bar in the fifth minute.
The visitors piled on the pressure with Mark Hudson drilling a powerful strike a yard wide in the 15th minute.
Hudson came close again in the 34th minute when he thundered a powerful strike straight at Dagenham keeper Tony Roberts from 30 yards.
United's hard work paid off in first-half stoppage time when Harrison poked the ball past Roberts from six yards after reacting quickest to a loose ball in the Daggers box.
The 24-year-old doubled his side's lead straight after the break when he volleyed home a Hudson cross from 20 yards in the 51st minute.
The home side tried to force their way back into the game but lacked a cutting edge with Dave Rainford summing up their poor form in front of goal with a mis-hit free-kick in the 66th minute.
Rotherham should have made it three in the 71st minute when Newsham connected with another Hudson cross only for Roberts to deny him with a spectacular one-handed save.
United had a penalty claim turned down in the 83rd minute when Ryan Taylor tumbled in the box under a Ross Smith challenge.
Newsham missed a chance to rub salt into the wound with six minutes left when he fired high and wide of an open goal from close range.