Jadon Sancho had put the Red Devils ahead in the 21st minute but the Saints found a way back with Che Adams’ 47th minute equalizer.
The result means United are fifth but have won only three of their past eight games.
“Unfortunately, again, it was very similar to the last couple of games,” Rangnick told reporters. “Very good first half, very good first half-hour.
“We did all the things we intended to do with the counter-attacks, with the deep runs. but then we stopped doing that. Even at the end of the first half, we didnt do those things anymore. And then in the second half, we also lost a bit of shape in the first 20 minutes.”
On why that happened, he added: “Again, we lost a little bit of composure, we made some wrong decisions, sometimes under pressure, sometimes not under pressure. This is where we have to get better, whenever we prepare the team for the next possible game.”
Elsewhere, Manchester City moved 12 points clear, albeit having played two more games than second-placed Liverpool, after a 4-0 win at lowly Norwich City.
Raheem Sterling netted a hat-trick for the champions, grabbing his third in the 90th minute when he swooped on the rebound after his own penalty was saved.
Frank Lampard’s Everton claimed an important 3-0 home win over Leeds United as both sides look to move clear of relegation trouble.
Brentford and Crystal Palace drew 0-0 while Brighton & Hove Albion won 2-0 at second-last Watford who are winless since November.