United manager Erik ten Hag named five players who had been at the World Cup in a strong starting line-up to face second-tier pacesetters Burnley, with Rashford, Bruno Fernandes, Casemiro, Eriksen and Tyrell Malacia all involved from the kick-off at Old Trafford.
Burnley, managed by former Manchester City captain Vincent Kompany, kept the game scoreless until the 27th minute when Denmark midfielder Eriksen struck from close range following a cut-back by Aaron Wan-Bissaka.
United's first game since the World Cup -- and their first since Cristiano Ronaldo's dramatic departure from Old Trafford -- saw them go ahead in the 27th minute through Denmark's Christian Eriksen, who also featured at Qatar 2022.
Rashford doubled United's lead just before the hour mark with a superb solo effort, the England forward running from inside his own half into the Burnley box before shooting into the bottom corner.
The England forward went on a run from inside his own half and into the visitors' box before drilling a shot low into the net.
'No time to feel sorry'
This was Rashford's first match since England's agonising World Cup quarter-final loss to France, with the forward saying: "It was such a disappointing feeling to be knocked out and you have to pick yourselves up.
"There is no time to feel sorry for yourself. You need to pick up results and thankfully we did that today."
United manager Erik ten Hag added: "There is still room for improvement but I am pleased with the win.
"The players are hungry after the World Cup break."
Brighton bowed out on spot-kicks three days after their Argentina star Alexis Mac Allister won the World Cup with Argentina in a penalty shoot-out.
Mac Allister was not involved on Wednesday after being given time off following his World Cup heroics and was thus spared any involvement at the south-east London home ground of League One Charlton.
A goalless draw saw the game go to penalties and after Addicks goalkeeper Ashley Maynard-Brewer saved from Moises Caicedo's penalty, it was left to Sam Lavelle to score the decisive goal as Charlton booked their place in the quarter-finals with a 4-3 shootout win.
Nottingham Forest, four-time winners of the League Cup, had no such problems against lower league opposition, the top-flight strugglers defeating Championship high-flyers Blackburn Rovers 4-1.
Forest went ahead at Ewood Park through Brennan Johnson's 13th-minute penalty before Scott Wharton made amends for conceding the spot-kick by equalising on the stroke of half-time.
But further goals from Jesse Lingard, Taiwo Awoniye and Johnson again took Forest into the quarter-finals for the first time since 1994.
