On an afternoon which saw goals in short supply across all matches in the Italian top flight, both sides had chances at the Atleti Azzurri d'Italia stadium but could not find the final touch.
Even Pinzi's red card after 63 minutes failed to help, with the point allowing Udinese to remain level on points with Juventus at the top of the table.
Antonio Di Natale fired a free-kick over the crossbar, while Pinzi also put an effort wide in a promising opening spell for the visitors.
Atalanta responded with Ezequiel Schelotto turning Giacomo Bonaventura's assist wide, before Simone Padoin was also off target with a strike from the edge of the area.
Chances continued to arrive at both ends of the pitch. Atalanta goalkeeper Andrea Consigli needed two attempts to gather a shot from Di Natale, and German Denis tried his luck from distance for the hosts but without any luck.
Schelotto failed to make the most of another opportunity eight minutes into the second half, instead driving wide.
Kwadwo Asamoah came even closer for Udinese, poking his shot from close range inches past the post before Di Natale soon followed with a volley which went high of the mark.
Things looked to be turning in Atalanta's favour when Pinzi's reckless challenge on Bonaventura earned him a second yellow card and an early shower.
That sparked a late spell of pressure from the hosts, but Friuli goalkeeper Samir Handanovic dealt superbly with everything that was thrown his way.
The Slovenian first tipped a volley from Simone Tiribocchi around the right post, and then palmed away a left-footed strike from Federico Peluso which had been helped on its way by a deflection from Medhi Benatia.
And Handanovic came to the rescue once more five minutes from time, punching clear a header from Tiribocchi which looked to be on its way into the top corner.