The Portuguese manager conceded his team have started better than last season but sees next month, when they face Liverpool and Tottenham Hotspur, as more significant.
"I know that last season (by) this fixture I think we would have, maybe, five or six points less than we have this season," he said. "So we are better this season than last.
"But this is nothing. It's just a start. A difficult period with an accumulation of matches, that comes in October. I really think that six teams are going to fight for the title."
Manchester City's 5-0 demolition of Crystal Palace on Saturday took them clear of United at the top but the hard-fought 1-0 win at Southampton showed off United's battling qualities instead of the attacking prowess they have demonstrated in recent weeks.
But United's victory was marred by more crude chanting from the away fans about goalscorer Romelu Lukaku.
United looked likely to win comfortably after Lukaku's 19th-minute goal, but had to rely on a hard-pressed defence in the second half and Mourinho praised Phil Jones and Eric Bailly's performance in the centre of the defence.
"And also Chris Smalling," he said. "We did for the last 20 minutes what the majority of the Premier League teams are doing, which is to play defensively with five at the back. We had chances to kill the game and score the second goal, but didn't."
The United boss, who was sent to the stands late in the match, said warm weather on the south coast had taken its toll on his players.
"The last time we had this was in Los Angeles and then in Macedonia, and I think they felt (it). I saw some players not sharp. Great spirit and always trying, but not the same sharpness. So if your opponent is coming with everything against you, let's do it for 10 or 15 minutes and stay solid."
He said he was baffled by his red card, which appeared to follow a collision with the fourth official and suggestions that he encroached on the field of play as Southampton pressed late on. "I don't know," he said. "Craig (Pawson, the referee) told me to leave, and I left."
Southampton boss Mauricio Pellegrino was pleased by his team's performance despite a ninth blank at St Mary's in 10 home games.
"To play with this spirit and with this football we will win more games than we lose and we have to continue in this way. They (United) realised that we were playing well and they defended. That is football too and you cannot always play really, really well. The best teams adapt to their situations."
United reiterated their condemnation of the crude chant about Lukaku after it was aired again during Saturday's match.
Both United and Belgian striker Lukaku had previously implored supporters to stop singing the song, which draws on racial stereotypes about the size of black men's penises.
But fans sang it again at St Mary's, as well as "We're Man United, we'll sing what we want", and the club now plan to use CCTV footage to identity the offenders.