Inter were frustrated at San Siro, following up just their second home league loss of the season on Sunday with another disappointing result.
They were slow to get going but dominated the last 30 minutes, with Danilo D'Ambrosio going closest when he had a late header cleared off the line.
Mazzarri told Sky Sport Italia: "I didn't like the first half, let me be honest. Udinese are a strong side who pressed us all over the field and closed down all the spaces.
"In the second half we deserved to score at least two or three goals, especially in the last half-hour. We tired Udinese out, dominated and then simply didn't score.
"At times there are errors in finishing, but at others there are elements we cannot control, such as the woodwork or the opposition goalkeeper. When we start playing the way we can, we create an incredible number of chances.
"In the second half I took Hernanes deeper, but wanted two against one on Udinese's left flank, using Jonathan and Ricky Alvarez. They hadn't predicted that and struggled.
"The important thing was to win and the team did everything possible to win. In my view they fully deserved a victory."
Udinese coach Francesco Guidolin admitted his side were under significant late pressure but was proud that they were able to hold on.
"The first half went as planned, only that we failed to score. It was what I expected and had hoped for," Guidolin said.
"During the second half we got deeper and deeper, had a moment in which I changed the tactical system because Inter had adjusted and they were pushing to the very end.
"From the hour mark onwards it was a very difficult game and we were suffering, though that was understandable against a side like Inter.
"We did well to defend, had a bit of luck and if we had always shown that attitude then I think we’d now have four or five more points that could've allowed us to look at Europe."
