Boro's hopes of securing a cupset were dented early on as David Silva put the visitors ahead inside three minutes. The hosts fell further behind as Sergio Aguero struck on 67 minutes and the two goal advantage was enough for City to advance.
Arsenal hosted National League giant-killers Lincoln City with the Imps hoping to add to a miserable week for the Gunners. However, this was not to be as Theo Walcott put the hosts ahead just before the interval with Oliver Giroud doubling the lead ahead of an own goal from Luke Waterfall for 3-0.
The rout and a place in the last four was wrapped up by two goals in three minutes as Alexis Sanchez and Aaron Ramsey registered efforts for 5-0.
Dele Alli and Vincent Janssen added efforts on 72 and 79 minutes before Heung-Min completed his hat-trick and a 6-0 hammering in added time.
The final game of the round was the biggest as PL leaders Chelsea hosted cup holders Manchester United on Monday night.
The visitors suffered a blow with the dismissal of Ander Herrera ten minutes before half-time and it took just six minutes of the second half for the Blues to take advantage as N'Golo Kante struck what would prove to be the winning goal in a 1-0 success.