The Blues had been hoping to negotiate a quarter-final clash with Bournemouth on Wednesday without the need to introduce their talismanic forward.
Hazard was, however, summoned from the bench in the 61st minute, with Barkley the man to make way.
The England international watched on from the sidelines as a player he considers to be one of the finest on the planet snatched a priceless match-winner six minutes from time to set up a semi-final date with London rivals Tottenham.
Barkley told Sky Sports of Hazard’s efforts: "The way we were playing, you could see we had the belief that a goal was going to come.
"When Eden got that goal, he can produce magic moments like that. Credit to him and the team who got stuck in.
"He's playing really well. He's up there with the best in the world. He can produce goals like that out of nothing and to play with a player like that is fantastic.
"He's relaxed, but once he's on the training pitch he's magical. He plays with enjoyment. He scores goals and creates goals. He's laid-back, but on game day he's ready to make things happen."
Hazard, who recently went 12 games without a goal for club and country, has now found the target in his last two appearances and has 10 club efforts for the season and 99 in total for Chelsea.
He will be looking to complete his century long before the Blues take in the first leg of their last four encounter with Spurs on January 8.
Barkley is, however, aware of how tough that two-legged contest promises to be, with Tottenham having already inflicted a 3-1 Premier League defeat on Mauruzio Sarri’s side this season.
The 25-year-old midfielder added: "Tottenham are a good side but we come into it with confidence and knowing what we need to do to reach the final. We will train really hard.
"That will be in the back of our minds [Chelsea's defeat at Wembley] because they are our rivals and we want to beat our rivals. It's going to be a very exciting game."