Twiss made it 15 league goals for the season with a superb goal on 28 minutes to keep Jim Harvey's side in the play-off race. The former Chester frontman took the ball on the left-hand side of the box and waltzed past several defenders before thumping a left-foot shot into the bottom right-hand corner of Tony Roberts' net.
It was the by far the best moment of a game that was more perspiration than inspiration with both sides, level before play on 47 points, working tirelessly but creating few clear cut chances.
Dagenham's best chance came after just 55 seconds of the game. Craig Mackail-Smith was presented with a golden opportunity eight yards out, but sliced his effort wide of the left-hand post.
Morecambe hit back with Garry Hunter forcing Daggers defender Scott Griffiths to make an important block, before Twiss gave the visitors a warning with a great run and shot on 12 minutes that was well saved by Roberts to his left.
The visitors retaliated with Glen Southam volleying wide from the edge of the area and Mackail-Smith having a shot blocked by Chris Blackburn.
The second half was one of few chances with Dagenham finding it difficult to break through a Morecambe defence once again well marshalled by Darran Kempson.
As John Still's team threw bodies forward gaps appeared in the Daggers defence and Morecambe almost capitalised.
Hunter clipped an effort just over the bar and Chris Blackburn came close with a diving header from Keiran Walmsley's corner.
Wayne Curtis came close to doubling the Shrimps' advantage in the second minute of injury time, but his shot was deflected wide as the home side ended the game on top.