The Reds were held to their seventh draw in 11 games at Anfield in 2011-12, as Stoke City ground out a goalless draw at the weekend.
Enrique, who played in Saturday's 0-0 Barclays Premier League clash despite a stomach upset, knows the team have to start winning matches on their own patch if they are going to qualify for next season's UEFA Champions League.
"We have to improve our home form, because we have lost too many points at Anfield this season," he said. "We just have to stay positive.
"I think, maybe the Swansea game aside [a 0-0 draw], we have deserved to win the other games we have drawn at home.
"If we start to take our chances then we can be in the top four, but if we don't then we will make it more difficult for ourselves."
On the road Liverpool's form has been far more impressive, and with a trip to relegation-strugglers Bolton Wanderers this coming Saturday, Enrique is hopeful Kenny Dalglish's men can return to winning ways this weekend.
"I think it's good we have a break now so we can prepare for Bolton," the 25-year-old Spaniard added.
"Away from home at the moment we are getting better results than at home, so hopefully we can get the three points next Saturday."