Inter had dropped to second in the table on Friday after Juventus' 1-0 victory over champions Napoli.
At the San Siro, midfielder Calhanoglu converted a penalty before setting up Federico Dimarco and Marcus Thuram scored a third for the 16-time champions before the break.
Argentina's Lautaro Martinez secured all three points with six minutes remaining to keep his spot as the league's top scorer.
"It's important to respond to Juve who are behind us, it's not easy to play after them," Martinez said.
"I'm proud and I hope to score many, the important thing is to help my teammates.
"I care little about who scores, the important thing is to continue like this," he added.
Simone Inzaghi's Inter are two points above Juve and remain unbeaten in Serie A and the Champions League since September 27.
Calhanoglu opened the scoring from the spot after 38 minutes following a foul on Martinez.
Inter led 3-0 at half-time after the 29-year-old delicately fed wing-back Dimarco to score and then Thuram netted his sixth league goal this season a minute before the break.
Thuram's effort came at the end of 21 passes in a move which started in the Inter half.
Udinese were denied a goal with 23 minutes remaining as on-loan Lorenzo Lucca was adjudged to be offside despite beating Yann Sommer.
Calhanoglu was replaced in the 71st minute by Albania defensive midfielder Kristjan Asllani before in-form Martinez completed the rout with six minutes to go.
Inter club captain Martinez tackled Udinese midfielder Martin Payero near the halfway line and charged forward to rifle his effort into the bottom corner for his 14th goal of the campaign.
Next up for Inzaghi's side is Tuesday's Champions League game with Real Sociedad, having already clinched their place in the last 16 of the competition.
Muriel magic
Earlier, Experienced Colombia striker Luis Muriel scored a 95th-minute winner with a backheel as Atalanta beat AC Milan.
The three points send Atalanta a point off the fourth and final Champions League place and stopped Milan from keeping up pace with the top two in the table, leaders Juve and city rivals Inter.
The defeat, just Milan's fourth in the league this season, comes at a bad time before Wednesday's crucial Champions League trip to Newcastle, where Stefano Pioli's side have to win to keep their hopes of reaching the knock-out stages alive.
Muriel's first goal of the campaign came after two efforts from Nigeria forward Ademola Lookman in Bergamo.
France attacker Olivier Giroud and Serbia forward Luka Jovic had equalised for the visitors.
Muriel, 32, claimed the winner, as he connected with Aleksey Miranchuk's drilled pass to secure Atalanta's first league win since October 30.
Sunday's highlight is Roma, in fourth, hosting sixth-placed Fiorentina.
