Southgate; I took over a mess
Gareth Southgate claims he has walked into 'a mess' after last nights goalless draw in Slovenia
Southgate replaced Allardyce last month, initially for four games, and led England to a home win over Malta before Tuesday's draw in Ljubljana.
England, who sit at the top of 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, 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 deafeat by underdogs Iceland.
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.
Southgate had dropped captain Wayne Rooney, but the Manchester United forward came on for the last 20 minutes.
Did he walk into a mess?
Chief football writer Phil McNulty: "There has to be a realism about what Southgate has walked into. He is in charge of a side that was made grimly aware of its place in the world order in France - and he played no part in that.
"Southgate did, however, have two eminently winnable matches to start with and, while four points keeps them top of Group F, the draw in Slovenia was the first qualifier of any kind England have not won in three years, halting a run of 14 successive victories going back to 2013.
"He also has a squad that has talent, certainly enough to be producing better than what was on offer on Tuesday.
"So yes, to a degree Southgate is correct to suggest he inherited a mess - but equally England have also been sorely lacking in fresh inspiration."