It means the Red Devils are eight points adrift of early pacesetters Arsenal, and still to record back-to-back wins for their new boss.
Ferguson was watching from the stands in his new role as a club director.
But Moyes insists the Scot's exit has not resulted in United's poor run of form.
"The fear comes from the team on the pitch," said Moyes.
"Obviously Sir Alex Ferguson has a great history and his experience will always work in charge of any team.
"But the players have always been the people that have to turn out and do it."
Unfortunately for Moyes, they are lacking the killer instinct at present.
United led through Robin van Persie and created countless other chances to seal victory.
But they failed to take their opportunities, leaving the visitors to claim a point after Lallana got the faintest of touches to Dejan Lovren's goal-bound effort.
"We should probably have put the game to bed earlier," said Moyes.
"We had a couple of chances, hit the bar and had a couple of other opportunities.
"We had some pockets of play when we were very good and there were times when we were not as good.
"It was a mixed game at times, but we still had those chances that I was hoping we would finish off."
The one bright spot to the day was the performance of Adnan Januzaj.
After signing a five-year contract earlier in the day, the 18-year-old delivered a brilliant pass to send Wayne Rooney clean through.
Although Rooney's effort was saved by Artur Boruc, Van Persie tucked home the rebound for the 26th-minute breakthrough, his first Premier League goal in open play since the victory at Swansea two months ago.
Januzaj brought a superb save out of Boruc with a powerful stinging effort and then released Danny Welbeck for what looked certain to be United's second before Morgan Schneiderlin came across to prevent the striker setting up England team-mate Wayne Rooney with a tap-in.
Moyes said of Januzaj: "We have been trying to get him signed for quite a few months, not just in my time.
"He's an exceptionally talented player and one we are happy to have.
"It was a great pass for the goal.
"I've said it a couple of times, but he's been worthy of his place.
"He has not looked out of place in the team. He has created goals and been part of of some of our best passing movements.
"He's a very talented player and we will try to use him at the right times and make sure we get the best out of him."
It hardly puts United in the best of moods ahead of their Champions League encounter with Real Sociedad on Wednesday.
However, for Southampton it represented another excellent effort after their recent victory at Liverpool.
"Getting a good win at Anfield and a positive draw today is amazing for the club," said Saints boss Mauricio Pochettino.
"We want to keep on believing we can do well. We set no limits. The sky is the limit."
Pochettino said he could not decide who to give his team's goal to.
"I'm not sure. I would give 50 per cent to Lovren and 50 per cent to Lallana. I haven't had a chance to see it again yet," he added.
Saints are riding high amongst the top six, maintaining their own advantage over a United side Pochettino admits are in transition.
"It's strange to come to Old Trafford and not see Alex Ferguson in the dug-out," he said.
"It's not easy for Moyes to be under the shadow of Alex Ferguson.
"But I think he is a great manager. He is a person I admire.
"I wish him all the best and hope he can be successful like Sir Alex."
- Sports & Recreation
- Premier League
- Adam Lallana
- Alex Ferguson
- Wayne Rooney