Arsenal had to overcome early nerves against Manchester United but the 2-0 win was reward for their patience and eventual sharpness in front of goal, manager Arsene Wenger said.
Danny Welbeck, Aaron Ramsey and Alexis Sanchez all went close before Granit Xhaka broke the deadlock in the 54th minute with a shot that looped up off Ander Herrera.
Welbeck then added a second with a fierce header after good work from Alex Oxlade-Chamberlain on the right.
"That's the kind of goal you want from Danny," said Wenger. "It was a great goal. I am happy for him. He has all the abilities a striker needs. Hopefully, that will give him a boost."
Although he did not respond to suggestions that Welbeck always does well against United, the striker was clearly delighted to score against the club where he spent 13 years.
"I grew up there and everything so it's a special place in my life," said Welbeck.
"But once we get on that pitch, it's business. When you score you've got to be happy, haven't you?"
The goal was his fourth of an injury-affected season and, with a little more luck, he could have scored two more in the first half. Twice he was denied by last-ditch defensive blocks as Arsenal kept United under pressure.
A breakthrough was always likely but Wenger was delighted that they kept the belief that it would come.
"It was a patiently built win where in the first half you could see we were a bit nervous, in the second half we took the edge off and scored the goals," he told SkySports. "It was a good win."
Wenger believes their season is gaining belated momentum after victories over Manchester City, Leicester City and now Manchester United.
Once again they are putting together a late-season dash up the table, with a top-four finish still possible.
"Mathematically it is still possible but now we need some help from the teams around us," said Wenger, who said he watched Liverpool's goalless draw earlier in the day.
"We had a bit of luck today with the first goal. We started a little nervously but slowly got better."
Jose Mourinho admitted that the defeat to Arsenal means it will now be "impossible" for Manchester United to finish in the top four.
United remain four points adrift of fourth-placed Manchester City with three games remaining, including a trip to second-placed Tottenham Hotspur.
Mourinho said their cramped schedule, which includes a possible two further Europa League games, means they must win the tournament to qualify for next season's Champions League.
"It is impossible to finish in the top four. We must play Thursday [against Celta Vigo] and then we go again against Tottenham, but we cannot go with everything against Tottenham," the United manager said.
"Three days later we have Southampton, and then Crystal Palace. Others will be playing once a week."
Mourinho said he was delighted with the performance of his players and said, under the burden of so many injuries and fixtures, he was surprised they managed to go 25 games unbeaten in the Premier League.
He said his task now was to continue to rotate his squad to ensure United win the Europa League. At the Emirates, Mourinho gave a Premier League debut to Axel Tuanzebe, saying he was delighted by the 19-year-old's performance.
"The kid was brilliant," said Mourinho, who asked him to play at right-back instead of his normal position in the centre of defence.
"Tuanzebe is a good player. He is a good kid and a good professional and it doesn't matter what position he starts. Every minute he spends with us training is fantastic for him, every minute in the Premier League is a plus.
"To play against Sanchez, [Mesut] Ozil, Welbeck ... these are big players. His best position is central defence but against the big boys it is right-back."
Reuters