United's top four chances already looked perilous before the game and the defeat left them in sixth place, six points behind fourth placed Arsenal, who also have a game in hand.
Nuno Tavares, Bukayo Saka and Granit Xhaka all got on the score sheet for Arsenal while Cristiano Ronaldo brought the score to 2-1 with his 100th Premier League goal for Manchester United, who also missed a penalty through Bruno Fernandes.
After losing 4-0 to Liverpool in midweek and getting the news of Erik Ten Hag's appointment as the club's new manager, this was a much improved performance from the Red Devils but they still left the Emirates Stadium empty handed.
Asked if the top four was out of reach for United, Rangnick told BT Sport: "Pretty sure. For me, even before the game it wasn't likely, but after today's result, the top four is gone, yes."
"There is a lot of work for sure [for Ten Hag], we knew before the game," Rangnick continued.
"But we also showed what kind of football we can play, and although we had to deal with the two early goals, we still showed the attitudes.
"So there is nothing about the attitude of the players that we should fault today."
The visitors felt hard done by after a handball by Cedric Soares went unpunished and Ronaldo had a goal chalked off by VAR while they was a hint of offside in Arsenal's third goal.
"The only weak side of the performance was we didn't defend well in and around the box," he said. "The same happened when we conceded the second and third goals, but for me there were three very unlucky, let's put it that way, VAR decisions.
"For me the third goal of Arsenal was clearly offside, you can see it in slow-mo. David said to me he couldn't see the ball.
"The second goal of Cristiano was definitely not offside and there was another handball decision in the first half, so we were not very happy with those VAR decisions in the end. It was an improved performance but disappointing result."
