Murphy reacted sharply to latch on to Zavon Hines' low cross to rob the home side of a fifth successive victory with a close-range finish.
Dagenham took the lead in the 20th minute through skipper Abo Ogogo, with Afolabi Obafemi claiming the assist with a perfect through ball for the onrushing midfielder to take in his stride and place a low shot under onrushing keeper Barry Roche from 12 yards out.
The Shrimps hit back seven minutes into the second half with Padraig Amond claiming his third goal of the season. Ryan Williams swung in a corner from the left-hand side and Amond nipped in front of his marker to flick a close-range header past the stranded Chris Lewington.
Jim Bentley's side then looked set for the full three points when Morecambe skipper Mark Hughes scored his first goal for the club on 76 minutes. Again a Williams dead ball caused the Dagenham defence problems.
The goal looked set to give the Shrimps another victory but Murphy came up with a leveller with the final kick of the game.