Higuain's 30th league goal of the campaign proved inconsequential as Bruno Fernandes' first-half brace was added to by Cyril Thereau before the Argentinian striker saw red 25 minutes from time.
Higuain had to be restrained by several team-mates following his second yellow card as he put his hands on the referee and tried to confront Felipe, who was the recipient of the kick which earned the striker his marching orders, before he shoved a member of Napoli's staff as he went down the tunnel.
The defeat was the fourth of the campaign for Maurizio Sarri's side and leaves them six points behind Juventus with seven games left as they chase a first Scudetto since 1990.
Udinese, who move six points clear of the relegation zone, took the lead in the 14th minute when Fernandes rolled the penalty under goalkeeper Gabriel after Kalidou Koulibaly had fouled Emmanuel Badu.
The lead lasted 10 minutes until Higuain broke Edinson Cavani's club record for most goals in a Serie A campaign by running on to a loose ball and rifling home a drive from the edge of the area.
Less than a minute later Fernandes missed the chance to double his tally as his second spot-kick was saved by Gabriel but he did put Udine back in the lead moments before the break with an overhead kick from just inside the area into an empty net.
The hosts made it 3-1 in the 57th minute when Thereau turned in Silvan Widmer's cross from close range before Higuain's dismissal and reaction rounded off Napoli's miserable day.