Barca booked their place in the Champions League semi-finals with a 3-0 win over United at Camp Nou, Coutinho completing the scoring with a fine 25-yard strike after Lionel Messi's first-half double.
The Brazil international, who has been criticised for a lack of consistency this season, celebrated his goal by facing the fans, putting his fingers in his ears and closing his eyes.
Rivaldo, who won two La Liga titles with Barca between 1997 and 2002, warned Coutinho against such antics and suggested he let his performances do the talking instead.
"It was not a good gesture. He scored a great goal, but it's never good to do something like that to the fans," Rivaldo told Marca .
"The supporters criticise him because they know he has to give more. I don't know what happened to him during the week, what he read in the newspapers or saw on television, but you have to keep working, think about the next game and score goals.
"If he scores everything will change and he will be a player for the club."
Coutinho arrived from Liverpool in January 2018 in a move that could reportedly cost up to £142 million and Rivaldo urged his compatriot to keep plugging away at Camp Nou.
"I would stay. He is a great player who cannot leave without being a Barcelona success because things haven't gone well," he said.
"Playing in Barca is impressive, they're in the Champions League, everyone talks about this team. You will not find another club like it in the world."
Rivaldo is confident Coutinho can get the fans onside by contributing more goals and rising to the occasion when Messi and Luis Suarez fail to do so.
He added: "He was a very expensive signing. Barcelona paid a lot of money for him and he is still not an idol of the fans. He must have more personality to make a difference.
"On the day Messi and Suarez don't score, he has to score. He should score 25 goals a season and, in addition, score in important matches."