The Reds looked down and out when they trailed 2-0 after 54 minutes but Daniel Sturridge, a half-time substitute, pulled a goal back in the 57th minute on his Premier League debut for the club and the visitors finished the game strongly.
"We probably let them control it in the first half. We sat back too much and just let them have the ball too much," said Downing.
"The manager stressed at half-time that we needed to push on and put them under pressure more when they had the ball.
"Sturridge coming on maybe pushed us a bit further forward.
"We got support for Luis Suarez up there, who was probably getting frustrated before that.
"That's what Sturridge gives you. He can get in behind people, and he's got pace and power - you saw that with the way he took his goal.
"I thought in the second half our performance was really good. Certainly for the last 20 minutes we had them pegged right back.
"You could sense that their fans were getting edgy and Alex Ferguson was shouting a lot of instructions.
"We had them rattled a bit and created plenty of chances."