Mulgrew and his Hoops team-mates slumped to a 1-1 draw to the Saints in an SPL match at McDiarmid Park on Tuesday evening.
However, Neil Lennon's team is still no fewer than 19 points clear of nearest rivals Motherwell at the top of the league table.
They have a Champions League last 16 rematch with Juventus in Italy and a William Hill Scottish Cup quarter-final with St Mirren still to look forward to.
And Celtic ace Mulgrew is confident that the defending Scottish champions are still capable of winning every game they are involved in.
He said: "We had been on a good run and had been winning games domestically. I don't think there was that much difference the other night.
"We are 19 points clear in the league so we are certainly putting it all together. We played well on Saturday and we also played well in midweek even though we did not win.
"We will have no problems putting the result behind us. We are looking forward to this weekend's game.
"Everybody is playing well at the moment. I think we have got a great squad here, whoever comes in does a great job and performs there role equally as well.
"It's a team effort at Celtic, we have shown that consistently. We win as a team and we lose as a team and I think that we will continue to do that."
Despite the 3-0 defeat they suffered in the first leg of the Champions League last 16 clash with Juventus, the Celtic players are looking forward to their rematch next month.
Mulgrew, though, knows the Hoops will have to focus firmly on their league matches and leave that showdown with the Serie A champions in Turin to the back of their minds.
Lennon's side take on bottom-placed Dundee at Celtic Park on Sunday in what will be their opponents first match since Barry Smith departed this week.
And the Scottish Player of the Year has urged his team mates to concentrate on picking up all three points against the Dens Park club this weekend.
He said: "Yes, the game against Juventus is something to look forward to. But we need to put it to the back of our minds and look forward to the next game that comes along.
"Our priority has to be the game with Dundee this weekend and with winning and picking up another three points in the league."
Meanwhile, defender Efe Ambrose will enter into talks with Celtic after reported interest from England, says his agent.
The Nigerian stopper has responded superbly to his Champions League calamitous performance against Juve by netting in two successive games against Dundee United and St Johnstone.
He's been linked with a move to the Barclays Premier League, with Sunderland, Swansea and Liverpool all credited with an interest.
And agent Dudu Dahan has revealed he will seek talks with the Parkhead club before making any further moves.
He said: "We are not thinking and not doing anything before we speak to Celtic first. Efe is under contract and is playing for one of the best clubs in the world."