Burnett was disappointed Dagenham failed to make more of their extra man, stating: "I'm very pleased with the result but I'm not overly pleased with the performance out there.
"It's going to be difficult against 10 men because they get behind the ball but in the second half they had better chances than us.
"I think we can play a lot better and I let them know that because we didn't have any penetration at all.
"The disappointing thing for me was we were inviting pressure and we have to be a lot braver on the ball.
"However, I'm not going to criticise them much because they are young and they're still learning and you make mistakes when you're still learning the game. Overall, I am also very happy to come away with three points."
Dagenham also triumphed at Roots Hall in the Johnstone's Paint Trophy just a fortnight ago, winning 5-2 and they were helped on their way this time by a red card to Phillips after just 27 minutes.
The Shrimpers centre-back was ordered off after a reckless challenge on Murphy who, just four minutes later, scored the game's winning goal as he coolly fired past the advancing Daniel Bentley.
After the break, Murphy missed a good chance to double Daggers' lead before Medy Elito volleyed inches wide of the target.
But the Shrimpers fought back well with Cauley Woodrow having a scorching 30-yard shot tipped over by Dagenham goalkeeper Chris Lewington.
The shot-stopper then also did well to parry a powerful Freddy Eastwood shot around the right post and Southend boss Phil Brown was pleased by his side's fighting spirit after issuing a strong team talk at the break.
"I really got in to the players at half-time, more than I have done before here, and they responded to that," said Brown.
"The first half wasn't good enough so we're looking at that rather than blaming the referee.
"We had a mountain to climb after the red card but the players wore the shirt with pride. They worked hard and couldn't really have given any more.
"There were plenty of positives for us in that second half and we could have got something from the game as their goalkeeper made a couple of very good saves, especially the one from Cauley Woodrow."