The away side went ahead after 25 minutes when Gabriel Magalhaes tapped in a knock-down from a corner.
Granit Xhaka and Thomas Partey both picked up yellow cards in a feisty match, which boiled over when Martinelli was booked twice in quick succession in the 69th minute.
The 20-year-old pushed Daniel Podence as he tried to take a quick throw-in then chased down Chiquinho to barge him over, prompting Michael Oliver to dismiss him.
Arsenal held on to earn a huge three points which moved them up to fifth in the Premier League, a point behind West Ham with two games in hand, but Morgan was left concerned with some elements of their performance.
Arteta was happy with the victory, but conceded that his side’s disciplinary record is becoming a worrying trend.
“To win football matches like that is extremely difficult and we have to stop it,” he told BT Sport.
“We’ve been talking about it. I have run out of ideas [about how to stop it], I think.”
He added: “The character, how we dug in, the spirit we showed, I’m really proud of the boys. There was togetherness and unity.
“We need to play with 11 players in the last 16 games and that is key. To win games with 10 men is very unlikely.”
Former Arsenal defender Martin Keown agreed with Morgan’s take on discipline, but was encouraged by the way the Gunners ground out a much-needed result to end a run of five matches without a win in all competitions.
“I think without doubt (it’s an issue),” he said on BT Sport.
“If you look at the two midfielders (Xhaka and Partey) tonight, they were both booked very early in the game. They’ve both been sent off recently.
“The discipline has to improve because eventually it will catch you out. Tonight they got away with it. It’s about building on it.
“Tonight I thought they turned that anger, what they thought was an injustice with the decision (Martinelli’s red card) to turn it into three vital points.”
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