However, David Robertson's strike after 57 minutes ensured a tension-filled end to the game but the champions hung on to go back to the top of the SPL.
Levein told Setanta Sports 1: "We missed too many chances. You come somewhere like Celtic Park and you have to take your chances.
"In the second half apart from (Shunsuke) Nakamura's chance I didn't think we were under pressure.
"We had great opportunities but we didn't score, when you have them you have to score but we didn't."
Dundee United next go to Hearts who are one point ahead in third and Levein is targeting third in the table, saying: "I can't ask for any more than that.
"The important thing is to go to Tynecastle and do ourselves justice.
"That's the way I look at it. That is where we want to be every season. We'll handle it."
Goalscorer Loovens acknowledged the Glasgow side had made life difficult for themselves after letting the opposition back into the game at 2-0.
"We are happy with the three points," Loovens added on Setanta Sports 1.
"At times we played some great football but at 2-0 it should be be over - but we made it too hard for ourselves in the end.
"We knew we had to be strong in the end to defend the three points."
Rangers travel to Hibernian on Wednesday and Loovens admits Celtic are also dependent on their city rivals to drop points.
"We are depending on them to lose points and we have to make sure we win our next two games. The pressure is on them now," he added.