Mulgrew and his Hoops team-mates extended their run without a defeat in the SPFL Premiership to 23 games with a 4-0 win over Kilmarnock last night.
And the Scotland international said Celtic gaffer Lennon has ensured his side's standards of play have remained high throughout the 2013/14 campaign.
Mulgrew said: "The manager is keeping on top of us and making sure we are always on our toes so that is helping. We are trying to do the best we can. The manager is on our case all the time in training every day in training trying to make sure we don't get slack."
Mulgrew stressed the players have not talked about going through the entire season without losing a Premiership match. And he has urged his team-mates simply to focus on the match against St Mirren at Parkhead on Sunday.
He said: "We will just keep playing away and take each game as it comes. We are not trying to think too much about records. It is there in the background and sometimes it is hard to ignore. It is a cliche, but we are just taking one game at a time."