Sheffield United boss Chris Wilder told BBC Radio Sheffield:
"It was a tough evening for us. It's something I've not really experienced before going into a game 3-0 up.
"It's a fabulous night for all concerned.
"Our attitude right the way through since the Burnley game [when their automatic promotion hopes ended] has been spot on.
"It was important to get through the first 20 minutes, when I didn't think our press was great and it allowed them to cause us some nervy moments.
"They've given it a right go. They've got some talented young players and they've been stretched by injuries and haven't got the biggest group, so it was a tough night for them but they gave it a go until the end.
"It looks a pretty straightforward two games but it hasn't been. The biggest thing was getting the job done which we have."
Bristol City boss Liam Manning told BBC Radio Bristol:
"Naturally going into it after the first leg we knew how difficult it's going to be. It's such a difficult place to come but I'm hugely proud of the lads.
"I think we came and we had a go without having a bit of conviction or quality at the end of it between the boxes. Especially early on the first 15, 20 we came out and got in some great areas but lacked quality in the final bit - whether that's composure, conviction, we lacked that.
"I think probably the frustration is what leads to their set-pieces - great learning for the lads, we're a young group playing in pressurised games like this.
"If we take the lessons, we reflect and we're honest with ourselves and step up and take responsibility, it will only make the lads better."