The Reds were leading through Dirk Kuyt's first-half strike when substitute John Arne Riise headed into his own net in the fifth minute of added time.
It also kept Grant, under intense pressure from his critics, on course for a Champions League and Barclays Premier League double.
He said: "I do not concern myself with what people say, I am just doing my job to get results.
"And I know that 1-1 away from home in Europe is good result. It is a big step now towards our goal."
He added: "We have conceded a lot of late goals this season so it was good to see one go in our favour.
"I felt we deserved it. There could have been a couple of penalties for us and we controlled the game until Liverpool scored.
"Both goals came from mistakes, but I must also praise Petr Cech for his performance. He made some important saves, but we are used to that from him.
"But although the draw is good for us, we know it will be tough at Stamford Bridge. It is always difficult for teams to play at Anfield with the support Liverpool's fans create, but it can also be intimidating for teams who visit Stamford Bridge.
"We know this is a big step towards the Champions League final now, and we believe we can complete the task in the second leg."
