Manchester City goalkeeper Joe Hart admitted his side played poorly as they beat Swansea thanks to Kelechi Iheanacho's injury-time deflection.
Victory took Manuel Pellegrini's team to the peak of the Premier League, but they looked lethargic after a Champions League win on Tuesday.
Swansea, who sacked Garry Monk on Wednesday, had equalised on 90 minutes after Wilfried Bony's opener.
"It was a big win considering we were awful," Hart told BBC Sport.
"I thought Swansea played really well considering the difficult circumstances. When we play well, I am a bit of a bystander so it was good for me to be involved.
"It's a good sign to not always play particularly well and be top of the league, qualify as group winners in the Champions League and to be in the semi-finals of the League Cup.
"To a man, we can all improve."
Manchester City now lead Leicester on goal difference, with the Foxes facing Chelsea on Monday, and could also be overtaken by Arsenal, who travel to Aston Villa on Sunday.
City travel to Emirates Stadium on 21 December for a crucial encounter with the Gunners.
Players did not recover from Euro win
After winning courtesy of Yaya Toure's shot, which took a big deflection off team-mate Iheanacho's back, Pellegrini said the result was more important than the manner in which it was achieved.
And he said several of his players were not fit enough to play the full game following their Champions League win over Borussia Mochengladbach on Tuesday.
Kevin de Bruyne came on as a substitute despite being Manchester City's joint top scorer with Sergio Aguero, who is currently out injured.
"We knew it would be difficult because we had three or four players who didn't recover from the last game," Pellegrini said.
"Kevin couldn't play the whole game, David Silva was able to play just 60 minutes, maybe if we finished 1-0 it would have been a normal game, the two minutes at the end is not the way we want to do it.
"Now we have a week to recover [for Arsenal], we have played too many games with 13 or 14 players."
Curtis proud despite defeat
Swansea caretaker boss Alan Curtis said he was proud of his team's performance despite failing to end a poor run, which now stands at one win in 12 games.
The visitors equalised in the 90th minute through Bafetimbi Gomis and Curtis contended that disallowing Gylfi Sigurdsson's goal was the wrong decision as Gomis was penalised for a foul on Hart.
But he was adamant that his team can climb the table, with the Swans a point above the relegation zone and poised to appoint a new manager before their home game against West Ham on 20 December.
AEK Athens boss Gus Poyet is among those linked with the vacant manager's post.
"We certainly deserved to get something out of the game, and when we equalised I thought we would hang in there," Curtis told BBC Sport. "We probably did create more chances than City, that's the difference between the teams at top of the league and at the bottom.
"We have been struggling with our form, but that's the standard we have to set between now and the end of the season.
"We are disappointed but we can look back with a little bit of pride in the way the team performed. We have 100% belief that we can get out of trouble."