After Odion Ighalo broke the deadlock, the Hornets rode their luck when Bradley Dack missed a penalty at the other end. But they did not heed their warning, as Mark Byrne sent the tie to extra time and Dack made amends with a winner for the League One club.
Edinburgh said: "In the end we deserved it. We had to ride our luck a little bit in the first half. After Saturday we were a little fragile.
"When we missed the penalty I thought we missed our chance but we showed great courage and spirit to keep going.
"For Bradley Dack to come back from the penalty miss shows the character of the man.
"Hopefully we can keep hold of him. We're not in a position to sell but if he does want to leave we'll get the right fee that his talents demand.
"It's a proud moment for the football club to come to a Premier League team with international stars and win. The level of fitness from my players was quite extraordinary.
"It's a credit to the players that they continued to press late on when Watford looked a bit jaded. It's without a doubt the biggest win of my managerial career and it's great for the club.
"We know that we had to fight harder and run harder than we usually do. We showed great tenacity in this respect and I'm so pleased for the players."