Sam Greenwood converted a first-half penalty for the away side, but City drew themselves level thanks to a delicate chip from winger Kayky early in the second half.
The night suddenly got tougher for City, as within a minute, midfielder Romeo Lavia had been sent off, and Greenwood had retaken the lead for the Yorkshire side.
Barry-Murphy was left delighted by the character shown by his side against a tough opposition featuring regular Premier League starters such as Luke Ayling and England international Patrick Bamford.
The head coach said: "I thought from start to finish we were excellent.
"The lads have had a tough month, some tough results and some games that have really tested the mentality of the group.
"The attitude and the way they kept going at all stages was the most satisfying thing for me.
"I’ve known Patrick since he was a very young player. His movement and the way he can exploit space in the box is second to none.
"It was a brilliant exercise for Luke Mbete and CJ Egan-Riley to see how they dealt with that and to see them respond in that way against a top-class opponent was brilliant from them."
An injury-hit EDS have endured a tough run, bowing out of the Papa John's Trophy and the UEFA Youth League, whilst also not picking up a win in the Premier League 2 in our last four outings.
But Barry-Murphy believes that testing period will only enrich the development of the players, evidence of which that was seen on Monday.
"I think those games that can be deemed as tough can also be the most rewarding once you come out the other side of it," he explained.
"To see the lads play with so much personality against a tough team is very good from my point of view.
"We were very creative in different situations and we created multiple chances and it would have been easy for us to feel sorry when we went behind with a sense of deja vu, or a sense of here we go again conceding at home.
"For so many parts of the season, our home stadium has been a fortress and we have played some great football so to see us express that in our play was very rewarding for us all."