Dusan Basta and Bruno Fernandes gave Francesco Guidolin's side a two-goal lead, but Pape Moussa Konate's first goal in Serie A reduced the arrears on the stroke of half-time.
Luis Muriel restored Udinese's two-goal advantage from the penalty spot just after half-time, but two goals from Alberto Gilardino earned Genoa a point.
There were chances at both ends early on, hinting at what was to come, with Fernandes denied by Mattia Perin and Marco Motta heading just wide for Genoa.
Roberto Pereyra wasted the best chance of the game come the midway stage of the first half when he was denied in a one-on-one situation by Nicolas Burdisso, who got back from nowhere and slid in to clear the danger.
Udinese took the lead 10 minutes before half-time when Emmanuel Badu's deep cross was met by the completely unmarked Basta, who volleyed in from six yards out.
Fernandes added a second before half-time when Pereyra beat the offside trap to bear down on Perin.
His shot was saved by the Genoa goalkeeper with the ball squirming out to Fernandes to meet at the far post.
Perin prevented further damage with a save from Fernandes before Genoa grabbed a lifeline just before half-time.
Giovanni Marchese's cross from the left was met by Konate, who placed a powerful header in from eight yards out, scoring his first Serie A goal in style.
The goal before the break was wiped out by one straight after it by Muriel, who converted from the penalty spot after Nicolas Burdisso fouled Silvan Widmer inside the area.
Genoa coach Gian Piero Gasperini let his frustration get the better of him and was sent away from his technical away by the referee before the hosts clawed their way back into the game for the second time through Gilardino.
Francesco Matuzalem dribbled his way out of danger in midfield and fed the Italian, who found the right-hand side corner from theedge of the penalty area.
Ten minutes later, Gilardino brought Genoa level when he was given a gift by Simone Scuffet, who flapped at a shot from Sebastian De Maio and laid the ball on a plate for the Rossoblu forward.
Genoa threw caution to the wind and searched for a winner, but Scuffet was not going to be beaten again after that mistake and the Zebrette held on for a point after seeming to be on course for three.