The result heaps more pressure on United and their management after sources told ESPN that the players are becoming frustrated with the training methods.
United remain in fifth place in the league table while Southampton stay in 10th position.
"We are conceding fewer goals than before but it doesn't help when you don't score more than one," Rangnick said. "We should have been two or three goals up after 20 minutes again.
"Then after two minutes [of the second half] exactly what happened at Burnley happened again, we conceded early in second half after being in possession. It is very frustrating for me and the players. It is a major problem, along with our goals."
The hosts did take the lead after 21 minutes after a fine move which saw Marcus Rashford set up Sancho for a tap-in at the far post.
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The visitors grabbed an equaliser three minutes into the second half through Adams' clinical finish from a tight angle.
Southampton also had chances to go ahead as an unmarked Stuart Armstrong wasted a golden chance and blazed the ball over the bar while Armando Broja also failed to find the target during a one-on one situation.
Ronaldo thought he had ended his goal drought after 70 minutes but his close-range header was disallowed for offside.
Veteran forward Ronaldo has now failed to score in any of his last six appearances in all competitions. The last time he had a longer run without a goal at club level was a run of seven games in December 2008 and January 2009.
Harry Maguire thought he grabbed a late winner but his header was brilliantly saved by Forster as the visitors held on for a point.
"Every minute we were getting better and played like we want," Saint boss Ralph Hasenhuttl said.
"We dominated in the second half and created super chances to score. We take a point -- not losing against Man City, Tottenham and Man United in our last three league games. We are not so bad."
The final whistle was greeted by boos from all around as interim coach Rangnick struggles to turn things around.
United have averaged fewer goals per game under the German than they have under any other manager in Premier League history.
Information from Reuters was included in this report.