Antony opened the scoring in the 35th minute following his move from Ajax before Bukayo Saka responded for Arsenal, but Rashford's brace ensured Erik ten Hag's side continued their early-season revival.
Mikel Arteta's team had won their opening five Premier League matches, but United halted Arsenal's perfect start to record their fourth successive victory after their initial struggles this season.
Arsenal looked to have the perfect start in the 12th minute when Gabriel Martinelli raced through and slotted home after being sent through by Saka, but the goal was disallowed after a VAR check for Martin Odegaard's foul on Christian Eriksen.
It was United who took the lead in the 35th minute through Antony when Rashford laid off for the Brazil international to fire past Aaron Ramsdale -- becoming the 21st United player to score on their Premier League debut.
Rashford had his second in the 75th minute when Eriksen squared for the England international to score from point-blank range, with the effort confirmed as being onside.
United look to have found their belief again after starting the campaign with dreadful defeats against Brighton and Brentford, with Ten Hag's approach now bringing results.
Antony was handed a place on the right flank with Rashford central and Jadon Sancho on the left, while Cristiano Ronaldo started on the bench alongside Brazilian Casemiro.
"The spirit from this team, they can deal with setbacks so we did, it is really great and shows your mentality. We have really improved on that," Ten Hag said after the match. "We are happy and satisfied with the win, also the performance against a really good team. They played really well."
Gunners boss Arteta was concerned his side had not made the most of a good performance.
"I am very disappointed that we haven't won the game," Arteta said. "The game was there for the taking. We haven't won it because we lacked some discipline in some moments and we weren't ruthless enough in front of goal.
"I told the players at half-time that the game is there for the taking and if we play with more courage we win the game. It was 1-1 and then it should have been 2-1 [for us]. It is a big lesson if you want to win here you have to do everything so right."
Arsenal remained top of the table on 15 points thanks to their opening five wins, while United climbed to fifth in the standings, three points behind the Gunners.