Napoli, already qualified for the Europa League thanks to their Italian Cup victory, move back sixth ahead of AC Milan with four games to play.
Gennaro Gattuso's side are out of the running for the Champions League next season but are preparing to meet Barcelona in the second leg of their last 16 tie in this year's edition, with both sides locked at 1-1.
Rodrigo De Paul had put 16th-placed Udinese ahead after 22 minutes at the San Paolo Stadium before Arkadiusz Milik came off the bench to pull the hosts level nine minutes later.
Politano, on-loan from Inter Milan, blasted in the winner five minutes into injury time for his first goal for the southerners who bounced back after two consecutive draws.
Roma, in fifth, host second-placed Inter Milan later on Sunday, with Antonio Conte's side looking to move three points behind leaders Juventus, who host Lazio on Monday.
At the bottom of the table Brescia, in 19th, won 2-1 to condemn SPAL to a return to Serie B after two seasons in the top flight.
In another relegation dogfight Genoa beat Lecce 2-1 to move four points clear of the southerners who occupy the final relegation spot.
Genoa are four points behind Torino, who lost 2-0 at Fiorentina.