In an edgy affair at the Academy Stadium, City’s Undwer-18s were pushed all the way by the Londoners after Lewis Fiorini had fired the hosts into an 11th minute lead.
Young defender Luke Mbete produced a crucial first half block to deny Sylvester Jasper before goalkeeper Cieran Slicker pulled off an astounding 86th minute save to deny Jonathon Page from just four yards out.
City can now look forward to a quarter-final clash at home to Burnley where the manage says the players focus and application will need to be stepped up once again.
“It was similar in some respects to the last round against Aston Villa,” said Taylor.
“I was pleased with elements and I thought we started the game well, but we slipped into some bad habits in the second period.
“And it was our own doing where we invited the pressure, but Fulham are a good team with strength on the counter-attack and Cieran made an unbelievable second half save and Luke Mbete for a 16 year made a vital first half block for us.
“We just about held it together at the end.
“I’ve not been happy with our training application over the past couple of days and if we don’t value the application, we get what we got in the second half and on another night, you get dumped out.
“It’s a Cup competition and you have to be at it for every occasion focused as there a lot who have fallen already.
“Now we have a good opportunity again against Burnley. We seemed to be away a lot last season so to have four fixtures on the bounce at home is nice.
“However, we can’t underestimate Burnley. They have already beaten a good outfit in West Brom already and it will be another tough game.”