The Reds are now five points adrift of the table-topping Gunners and only a point behind Chelsea going into the international break. Manchester City sit further back in eighth place following their surprise reverse at Sunderland.
Cleverley came off the bench at half-time to replace Nemanja Vidic and played his part in a dogged performance that yielded a fine victory, courtesy of Robin van Persie's header.
"Going into the game and seeing the weekend's results, we knew it was an extra-important three points," said the England international. "We managed to grind it out in the end with a great defensive performance. We had some more chances to seal it.
"We didn't want Arsenal to go 11 points ahead of us. This gets us right back in the mix and gets our confidence up. This is a big win for us as we are ahead of City and not far behind Chelsea. We have also played some very tough games out of our opening 11 fixtures.
"I am sure we can get a run together and get right up to the top. It is what everyone predicted at the start of the season - everyone is improving and it makes for a really competitive season."