However, Lazio easily fended off Juve's lackluster attacking, as the only goal of the match came when Gila accidentally redirected Juan Cabal's cross into his own net, handing Juve the crucial three points.
"It was a difficult match against a good team," said Motta.
"It's our first win at home (in the league) since August and it was deserved."
Lazio, in fifth, were angered by Douglas Luiz not being sent off for punching Patric, an offence which was apparently not seen by the VAR officials.
"It is violent conduct and should have been a red card. I do not understand why it was not sanctioned. We feel that we are being taken for a ride," said sporting director Angelo Mariano Fabiani.
Juve have a better goal difference than Napoli but stay behind their rivals, coached by Antonio Conte, as they have played a game more.
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Napoli can ensure they stay top with a win at Empoli in Sunday's lunchtime match, while champions Inter Milan, who sit two points off the pace alongside their local rivals Milan, are at Roma.
Battling Milan
Samuel Chukwueze's 13th-minute strike looked like being the first of a comfortable win for Milan but ended up being the winning goal as the seven-time European champions held on for the points.
Their task was made difficult by Tijjani Reijnders being sent off in the 29th minute for a clumsy foul on Sandi Lovric, who would have been clean through on goal had the Netherlands midfielder not clipped his heels.
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"It was two matches in one. The first one finished after half an hour, 30 minutes of character, quality and the sort of play which is close to what I want," said Paulo Fonseca.
The Portuguese has had to field questions about ill discipline in the dressing room, especially among star players like suspended Theo Hernandez and Rafael Leao, who was left on the bench for the entire match.
"After the sending off it was all about the team spirit," he added.
"If anyone doubted that we're a united team today was the demonstration that we are."
The hosts looked to have been sucker-punched when Christian Kabasele bundled home for Udinese in the fifth minute of stoppage time, but a huge roar rang around the ground when, after a long VAR check, the goal was ruled out for Jurgen Ekkelenkamp's toes straying offside.
Ultra protests
Milan's win came in front of a flat San Siro following the decision of Milan's hardcore "ultras" supporters to stand in silence for most of the match in protest at what they claim is a law enforcement and Italian media campaign against them.
Leading ultras from both Milan and local rivals Inter were arrested last month and are accused of a variety of crimes ranging from criminal conspiracy and extortion to assault.
Earlier this week, Milan's ultras groups refuted allegations by investigators of any involvement in ticket touting, control of parking and sales from concession stands near the San Siro.
In the meantime, police banned the ultras' large "Curva Sud Milano" from being hung out in that section, leading to supporters around the whole stadium, not just ultras, leaving their own banners and flags at home.
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Instead fans in the Curva Sud held a banner in support of their arrested allies which read, loosely translated, "Stay strong boys".
(This story has not been edited by staff and is auto-generated from a syndicated feed.)