Massimiliano Allegri's first home game in charge of Juventus got off to the perfect start as Tevez scored with a confident finish after good work from Stephan Lichtsteiner in the eighth minute.
Despite being without injured Chilean midfielder Arturo Vidal, Juve controlled the game well without creating many clear-cut chances.
Udinese thought they had drawn level in the 66th minute when Igor Bubnjic had the ball in the net but the Croatian was ruled offside.
Tevez went close again with a fierce drive that brought a fine save out of Greek keeper Orestis Karnezis but Juve finally got their second through a superb 25-metre strike from Marchisio.
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Roma's win at Empoli arrived courtesy of an own goal on the stroke of halftime.
Impressive Belgian midfielder Radja Nainggolan's low drive from 20 metres struck the post and bounced into the net off the back of keeper Luigi Sepe.
Roma coach Rudi Garcia fielded a slightly weakened side ahead of Wednesday's Champions League game at home to CSKA Moscow.
Skipper Francesco Totti and Ivory Coast winger Gervinho were rested and while Roma missed their creative impact, they were still comfortably in charge for most of the game.
Shortly before the goal Maicon hit the post from close range and after the break Sepe foiled the Brazilian defender again.
Empoli coach Maurizio Sarri was shown a red card in the 72nd minute after protesting as he felt a penalty should have been given when Levan Mchedlidze went down in the area.
Five minutes later Georgian Mchedlidze forced a fine save out of Morgan De Sanctis as Empoli pushed for an equaliser.
Roma and Juve have six points from two games while Udinese are on three and Empoli have zero points.
Sunday's games see Parma host AC Milan and Napoli take on Chievo Verona.
(Reporting by Simon Evans, editing by Tony Jimenez)
