Southgate succeeded Sam Allardyce last month, initially for four games, and led England to a home win over Malta before Tuesday's draw in Ljubljana.
England, who top Group F, had keeper Joe Hart to thank for a point.
"With the overall objective of qualifying, we have kept the team on track," said Southgate.
"But we have taken over a mess really and had to steady the ship."
The draw means England remain top of the group with seven points, two more than Lithuania and Slovenia
Southgate, 46, stepped up from his role as England Under-21 boss to take temporary control of the senior team following Allardyce's departure.
His next game is a World Cup qualifier against Scotland at Wembley on 11 November, followed four days later by a friendly with Spain, as the Football Association searches for a permanent successor.
Allardyce, 61, left his post after one game following a Daily Telegraph investigation claiming he offered advice on how to "get around" rules on player transfers.
That followed England's disastrous Euro 2016 campaign, which saw Roy Hodgson resign after an embarrassing last-16 defeat by minnows Iceland.
England have two wins and a draw from their three matches in this campaign, but the latest result ended a run of 14 straight qualifying victories going back to 2013.
Slovenia, ranked 67th in the world, had the best chances in the game, while England failed to register a shot on target in the first half and managed just three overall.
"The overall objective is to qualify, so long term that could be an important point," Southgate told ITV. "Everyone can see we could be better but their chances came from our mistakes or set-pieces."
Southgate had dropped captain Wayne Rooney, but the Manchester United forward came on for the last 20 minutes.
'We owe Hart for point'
Goalkeeper Hart was praised after he helped spare England embarrassment with a succession of outstanding saves, including a world-class reflex stop when he turned Jasmin Kurtic's header on to the crossbar seconds after half-time.
Hart, who is on loan at Italian side Torino from Manchester City, has struggled recently, making high-profile errors against Wales and Iceland at Euro 2016.
Southgate revealed he had a "long chat" with the 29-year-old earlier in the week.
"We owe our goalkeeper really for getting a point," added Southgate. "He was fantastic.
"He's at his best when he's calm and composed, it's something we've discussed and this week his manner has been very calm. He was at his best tonight.
"We didn't start either half well and it's an outstanding save that gets us a point."
Credit: BBC sport