Daggers came from two goals down to earn a share of the spoils in wet and windy Devon.
The visitors arrived at Plainmoor looking for a fifth successive league win, having started 2022 with a 3-1 home win over Dover Athletic.
Boss Daryl McMahon was back in the dugout, having missed Monday's match due to Covid protocols, but saw his side fall behind within a minute as Asa Hall headed past Elliot Justham.
The Daggers keeper saved from Stephen Wearne soon after, with the visitors going close to a leveller on 11 minutes through Junior Morias.
Justham produced a great save to deny Wearne moments later but was beaten for a second time as Danny Wright doubled Torquay's advantage.
McMahon's men slowly grew into the game, without being able to work many openings in the attacking third, until Morias got to the byline and cut the ball back for Callum Reynolds, who fired over on 21 minutes.
Josh Walker was just off-target four minutes later after more good work by Morias, but Daggers halved the deficit on the half-hour mark when Angelo Balanta headed a Myles Weston corner goalwards and the ball fell kindly for Walker to convert.
Weston then curled a free-kick just past the far post on 36 minutes and he was denied by Dean Moxey's goalline block, with appeals for handball waved away, before Will Wright curled a free-kick wide from 20 yards four minutes before the break.
But Daggers had a let-off three minutes after the restart as Danny Wright got through on goal and beat Justham, but saw the ball hit the side netting.
Morias forced a brilliant save from McDonald just two minutes later, though, after being played through at the other end, and Mo Sagaf collected the ball on the edge of the box and forced the Gulls keeper into another superb save to his left on 53 minutes as Daggers continued to dominate.
Weston was next to test McDonald, before McMahon sent Paul McCallum on for Walker.
But the hosts enjoyed a spell of possession and sight of goal midway through the half only for Ben Wynter to fire well over Justham's crossbar.
Balanta dragged a shot wide for Daggers on 70 minutes, but they were back on terms just moments later as their captain cut inside and curled a brilliant shot into the top corner.
And the visitors continued to press, with Wright seeing penalty appeals waved away after going down in the box and Morias pulling a shot inches wide from just inside the penalty area on 77 minutes.
Justham gathered safely from Dan Holman's downward header nine minutes from time, before Joey Jones found the excellent Sam Ling who tested McDonald.
And Balanta saw a curling effort blocked during three minutes of stoppage time as it ended all square and Daggers climbed up to fourth.
Dagenham: Justham, Wright, Reynolds, Zouma, Ling (Vilhete 89), Jones, Sagaf, Weston, Balanta, Morias, Walker (McCallum 64). Unused subs: Clark, Phipps, Akanbi.