Lautaro led the league with 24 goals last season and scored a Copa-best five goals - including the winner in the final - to lead Argentina to the trophy in July.
But he acknowledged after last weekend's derby loss to AC Milan that he hadn't been in top shape after going scoreless in Inter's opening five Serie A games - plus a 0-0 draw at Manchester City in the Champions League during which he was benched at the start for his lack of fitness.
"He's our captain and we all know what he represents," Inter coach Simone Inzaghi said. "He had always trained hard and today he was great. … Sometimes you bounce back stronger from defeats."
After Davide Frattesi opened the scoring for Inter and Christian Kabasele equalised for Udinese, Lautaro struck either side of halftime.
First, he bundled the ball over the line after missing an initial attempt. Then he finished off a setup from Marcus Thuram to revive the scoring tandem that powered Inter to the title last season.
Lautaro last scored in Serie A in March.
Lorenzo Lucca pulled one back for Udinese late in the second half.
Inter moved level on points with Torino and Milan, one point behind Juventus, who took the lead with a 3-0 win at Genoa. Milan beat Lecce 3-0 on Friday, while Torino host Lazio on Sunday.
Dusan Vlahovic scored twice as Juventus ended a run of three consecutive 0-0 draws with a 3-0 win at Genoa - where Juventus coach Thiago Motta began his managerial career.
Juventus have kept six consecutive clean sheets to start the season. One more clean sheet and the Bianconeri will match the record set by Cagliari in 1966.
The match was played without fans as punishment for violence between opposing supporters before and after the midweek Genoa derby.
Vlahovic converted a first-half penalty then took a pass from Teun Koopmeiners to score another from a tight angle.
Francisco Conceicao sealed it for Juve in the 89th by redirecting a cross from Khephren Thuram.
Lazar Samardzic scored in the 90th minute for Atalanta to draw with 10-man Bologna 1-1 in a matchup of Champions League clubs.
Bologna's 20-year-old striker Santiago Castro scored for the third consecutive game a minute into the second half with a shot from beyond the area.
Six minutes later, Bologna were reduced to 10 men when Jhon Lucumi was shown a red card for a foul on Charles De Ketelaere.
Atalanta kept pressing and finally equalised with a long, curling effort from Samardzic.
On Wednesday, Bologna visit Liverpool and Atalanta play Shakhtar Donetsk.
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