The Reds drew 1-1 with United at Old Trafford on Thursday night to secure a 3-1 aggregate win in their Europa League last-16 clash.
Liverpool are four points behind Louis van Gaal’s sixth-placed side in the table, although the Merseyside outfit have played one game less than their bitter rivals.
Jurgen Klopp’s men make the trip to Southampton at St Mary’s in the Premier League on Sunday afternoon, while United take on Manchester City in the derby at the Etihad Stadium.
Former United midfielder Hargreaves is backing Klopp’s side to finish above the Red Devils in the Premier League table.
“Liverpool will finish above Man Utd,” Hargreaves told BT Sport.
“They’re flying and the form that they’re in. It is great to see a philosophy and identity brought in by Jurgen Klopp.”
Liverpool have won three of their last four games in all competitions ahead of Sunday’s trip to St Mary’s.
The Reds will face Klopp’s former club Borussia Dortmund for a place in the Europa League last four.
MORE: The latest football news