United were joined in the fourth round by fellow Premier League sides Brentford, who beat third-tier Leyton Orient 3-1, and Crystal Palace who edged past Queen's Park Rangers 2-1.
Premier League Fulham were knocked out in extraordinary fashion by second-tier Preston North End after the match ended 1-1 -- Preston prevailing 16-15 on penalties, the longest-ever shootout in the competition.
England forward Rashford ended a six-month goal drought at the weekend when he scored against Southampton in the Premier League and he needed only 16 minutes to strike against Barnsley.
His silky finish paved the way for United to demolish Barnsley who were no match for Erik Ten Hag's side.
Antony added the second from the penalty spot in the 35th minute and Garnacho poked home United's third in first-half stoppage time to put the hosts in complete control.
Garnacho side-footed his second from Rashford's pass just after the restart and returned the compliment as Rashford made it 5-0. Eriksen scored his first United goal for more than a year in the 81st minute, slotting in Bruno Fernandes's cross.
Eriksen then fired in from the edge of the area to complete United's biggest-ever League Cup win.
The Premier League's bottom two clubs met at Goodison Park and there was more woe for Everton as they bowed out on penalties to Southampton after 1-1 draw.
Abdoulaye Doucoure put Everton ahead but Southampton levelled through Taylor Harwood-Bellis and the visitors won 6-5 in the shootout. Eleven penalties hit the net before Ashley Young saw his effort saved by Alex McCarthy.
Brentford were given an early scare when Brandon Cooper put Orient in front but Fabio Carvalho equalised in acrobatic fashion. Mikkel Damsgaard headed Brentford in front and Christian Norgaard ensured progress for Thomas Frank's team.
Eddie Nketiah scored his first goal for Crystal Palace and Eberechi Eze struck the winner against his old club QPR to send Palace into the last 16.
Sheffield Wednesday won 1-0 at Blackpool while Championship side Stoke City needed penalties to get past fourth-tier Fleetwood Town after being held to a 1-1 draw.