Daggers made two changes for the trip to Plainmoor, with Luke Howell replacing Andre Boucaud and Craig Robson coming in for Josh Staunton.
After early chances for Oliver Hawkins and Fejiri Okenabirhie, the Essex club had skipper Scott Doe sent off for a foul in the 15th minute.
The centre-back was judged to have pulled down Torquay’s Dan Sparkes as the last man, leaving referee Antony Coggins with little option but to send Doe off.
Eight minutes later, the Gulls took the lead when Jamie Reid fired in for the hosts to make the Victoria Road side’s task a lot harder.
Daggers almost equalised in the 35th minute from a corner, but were thwarted by a combination of Torquay midfield Damon Lathrope and keeper Brendan Moore.
However, the visitors could not find a goal and headed into the break behind with the score 1-0.
Neither side could fashion a clear cut chance in the opening exchanges of the second half, though Hawkins narrowly headed over in the 57th minute.
Substitute Chris Assombalonga almost levelled for Daggers in the final minute of stoppage time, but headed a Matt Robinson cross over the bar.
Daggers will hope to rebound swiftly when they make the trip to the Silverlake Stadium to face Eastleigh on Saturday.
Torquay: Moore; Ward, Galifuoco, McGinty, Rower-Turner; Richards, Lathrope, Young, Sparkes (Chaney 54); Blissett, Reid (Williams 73)
Subs: Chamberlain, Nicholson
Daggers: Justham; Widdowson, Doe, Robson, Raymond; Robinson, Howell (Ling 75); Maguire-Drew (Staunton 19), Whitely, Okenabirhie (Assombalonga 69); Hawkins
Subs: Cousins, Donnellan
Attendance: 1,638 (including 70 Daggers fans)
Referee: Antony Coggins