In normal circumstances, this victory and performance by Albion would not live very long in the memory.
The striker’s 20th goal of the campaign in the second half was enough to secure an eighth victory in succession, which is a post-war club record.
Murray now has the highest tally of his career and Albion are 13 points clear at the top of the table.
Five of their record-breaking eight wins have been 1-0 which is testimony as much as anything to a defence now unbreached away from home for more than seven hours.
Poyet named an attacking line-up by recalling Elliott Bennett and retaining Craig Noone following Saturday’s 2-1 home win against Swindon.
They have not started many games together since Noone arrived from Plymouth during the January transfer window.
Matt Sparrow dropped to the bench but Fran Sandaza kept his place upfront in preference to Chris Wood, who had a long journey back from China where he played in a friendly for New Zealand.
Dagenham, unchanged after Saturday’s 1-1 home draw against Sheffield Wednesday, were spearheaded by former Seagull Bas Savage.
During his spell with Albion Savage was best known for his Moonwalking goal celebration.
The amiable striker has not lost his taste for the unorthodox. His first contribution was to send a header in the wrong direction straight out of play, his second to accidentally get in the way of a fierce shot from team-mate Romain Vincelot.
Savage’s next involvement, in the 16th minute, very nearly saw Albion fall behind in an extraordinary train of events.
A clever piece of skill by Jon Nurse took him away from Gordon Greer for an angled drive which Casper Ankergren could only parry.
A combination of goal-line clearances by Tommy Elphick and Marcos Painter, together with the underside of the crossbar, foiled Savage’s attempts to convert the rebound.
It was a miraculous escape for Albion in circumstances which always promised to be testing.
They have coped admirably all season with the contrasting challenges which League One offers, from Southampton and the spacious St Mary’s stadium to Dagenham and their compact Victoria Road home.
The pitch was hardly ideal for their passing brand of football either. It was both firm and bumpy, which made judging the bounce of the ball awkward.
Dagenham, out of the relegation zone after a six-match unbeaten run, were awkward opponents too for Poyet’s runaway leaders.
Most of the action in the opening half hour was too close for comfort for the 1,500 Albion fans behind Ankergren’s net.
The Dane had to hold another low shot from Savage after he had got the better of Greer just outside the box.
There was plenty of attacking intent in Poyet’s formation, Noone playing high up the pitch on the left to effectively form a three-pronged forward line with Sandaza and Murray.
They had in behind them the creative talents of Bennett and Gary Dicker but some of the passing was not quite as precise as normal while, at the back, defenders who have customarily been composed this season looked a little ruffled on occasions.
All in all, Albion were well below their best in the first half. In fact, they did not manage a single shot on or off target, which is very rare for them.
Bennett had an effort of sorts from outside the area early in the second half which skewed high and wide of the target.
It was a good night for Albion’s chief provider to be under performing, given that Norwich manager Paul Lambert was in the crowd. The Canaries repeatedly tried to lure Bennett to Carrow Road in January.
Dagenham continued to carry the greater threat. Damian Scannell clipped a shot narrowly wide after good work by Nurse and Savage had another drive parried by Ankergren after he had once more got the better of Greer.
Even when Albion took the lead against the run of play just before the hour, the breakthrough had an element of fortune about it.
Noone was trying to shoot from 20 yards out but it turned into a cross for Murray, lurking at the far post, to volley in via the bar.
Albion are very difficult to peg back once they have their noses in front. Goals change games and they became more authoritative, although Dagenham remained intermittently dangerous.
An inviting cross from Danny Green lacked only a decisive touch from a team-mate then Murray, assisting his defence, blocked a well-struck shot by Peter Gain.
Southampton manager Nigel Adkins, accompanied by his assistant and former Albion centre-half Andy Crosby, had seen enough with 15 minutes left.
He departed to chants of “Adkins, Adkins, what’s the score?” from the Albion faithful.
Southampton, the hot pre-season favourites for promotion, are the visitors for the penultimate match at Withdean next month. There is every chance the Seagulls will have wrapped up not just a place in the Championship but also the title by then.
Albion (4-3-1-2): Ankergren; Calderon, Greer, Elphick, Painter; Bridcutt, Dicker (Sparrow 75), Noone (Wood, 84); Bennett (Kishishev 60); Murray, Sandaza. Subs not used: Brezovan, Navarro, Taricco, Dunk.
Scorers: Murray (59).
Yellow Cards: Calderon (21) foul.
Red Cards: None.
Dagenham and Redbridge (4-4-2): Roberts; Ogogo, Arber, Doe, McCory; Green, Vincelot, Gain, Scannell (Currie 63); Nurse, Savage (Lancaster 66). Subs not used: Lewington, Antwi, Ifil, Lee, Morgan.
Yellow Cards: Arber (69) foul, Ogogo (90) foul.
Red Cards: None.
Referee: Andy Woolmer (Northants).
Venue: Victoria Road.
Attendance: 3,604 (1,411 Albion).
Next match: Rochdale v Albion League One Saturday (3pm).
Entertainment value out of five: 1.