The Gunners had an effort cleared off the line and United owed a small debt to David De Gea for a handful of smart saves, while Robin van Persie twice went close to breaking the deadlock against his former club.
In the end, though, both teams had to settle for a point, even though Cleverley believes his side probably did enough to earn victory.
“I think it was a pretty fair result,” Cleverley told MUTV. “We feel like we could have won the game, but their goalkeeper made a couple of good saves. It could have swung either way but in the end it’s a fair result.
“We need to win every game from now on and Liverpool getting a last-minute winner [at Fulham] is disappointing for us. Our result isn’t the worst in the world but we need to start winning games fast.”
Cleverley admitted Robin van Persie could have given the Reds a dream start – the Dutchman blasted straight at Gunners goalkeeper Szczesny when gifted the ball by Mikel Arteta – but the midfielder wasn’t about to point fingers.
“Nobody’s blaming Robin for missing chances because he’s scored so many goals for us this year.