Bradley helps Roma set new record
Roma played the final 24 minutes with 10 men after Maicon was sent off for a second yellow card.
Juventus recorded eight opening victories in 1930-31 and 1985-86, and also won nine straight matches to start 2005-06 but was stripped of its title-winning season in the ''calciopoli'' match-fixing scandal.
''At the end it was three deserved points, we really gave everything,'' Bradley said. ''This is a team which has great desire and determination.
''Last year didn't go how we wanted it to, now it's going a bit better. We're working every day to improve and we're not thinking about the statistics.''
Elsewhere, Gonzalo Higuain converted two controversial penalties to help Napoli beat 10-man Torino 2-0, while Juventus beat Genoa by the same scoreline.
''Along with Napoli, we're on a run that befits a top of the table team,'' Juventus coach Antonio Conte said. ''But Roma's results skew everything a bit, their run is the very best.
''We scored two goals and missed so many other chances. But in a certain sense, the good news of today is that we didn't concede.''
Udinese was unbeaten in 22 home games and it almost got off to the perfect start when Luis Muriel's effort crashed off the left upright with less than three minutes on the clock.
Roma clearly missed talismanic captain Francesco Totti, who is out injured, and Udinese had another fantastic chance in the 38th minute when Gabriel Silva chipped goalkeeper Morgan De Sanctis, but Leandro Castan cleared off the line with an acrobatic overhead kick.
A Roma victory appeared even more unlikely when Maicon was given his marching orders in the 66th.
However, Bradley scored the winner minutes after being brought on for Marco Borriello. Kevin Strootman was given too much time on the ball and he spotted Bradley's run, allowing the U.S. international to drill the ball into the bottom left corner.
Juventus badly needed a win after a miserable week which saw Antonio Conte's side defeated at Real Madrid in the Champions League, days after letting slip a two-goal lead to lose 4-2 against Fiorentina.
Arturo Vidal, who had earlier hit the crossbar, fired Juventus in front from the spot in the 22nd minute after Davide Biondini tripped Kwadwo Asamoah, although the foul appeared to be just outside the area.
Juventus almost doubled its lead moments later but Genoa goalkeeper Mattia Perin pulled off a stunning save to deny Andrea Pirlo from close range.
The two-time defending champion did grab a second 10 minutes from halftime when Carlos Tevez dribbled in the area before placing an angled drive across into the bottom right corner.
In the lunchtime fixture, Napoli took the lead in the 14th minute when Nicola Bellomo was adjudged to have tripped Dries Mertens and Higuain blasted the resulting penalty under the bar.
Torino was again furious when Napoli was awarded another soft penalty shortly after the half hour mark when Federico Fernandez's shot came off Kamil Glik's arm from point blank range. Higuain again converted, firing into the bottom right corner.
Torino's misery was compounded when halftime substitute Migjen Basha was shown a straight red card four minutes from time after hauling down Goran Pandev, who was clear through on goal.
Milan was buoyed by a midweek draw against Barcelona but still beset by injury problems, missing Christian Abbiati, Stephan El Shaarawy, Giampaolo Pazzini and Daniele Bonera, while Philippe Mexes was suspended.
It found itself 2-0 down at halftime after poor defending allowed Marco Parolo and former Milan striker Antonio Cassano to score.
Milan coach Massimiliano Allegri brought on Alessandro Matri and Kaka for Mario Balotelli and Andrea Poli seven minutes into the second half and the changes appeared to work. Moments after Walter Gargano hit the crossbar, Matri scored his first goal since signing from Juventus in the offseason.
And Milan was level two minutes later when Matias Silvestre headed in a rebound after Cristian Zapata's attempt had come off the bar.
Milan thought it had earned a point but Parolo drilled a low free kick into the bottom left corner with the last kick off the match.
Parma has leapfrogged above Milan into eighth, two points behind Lazio which beat Cagliari 2-0 thanks to a header from halftime substitute Miroslav Klose and a penalty from Antonio Candreva.
Fiorentina moved fifth, nine points behind Roma, after coming from behind to win 2-1 at Chievo Verona thanks to a double from Juan Cuadrado. Chievo slipped to bottom spot as Jose Angel Crespo's third-minute header helped Bologna beat fellow struggler Livorno 1-0 for its first win of the season.
In another match between two relegation candidates, new Catania coach Luigi De Canio steered his side to a goalless draw at home to Sassuolo on his debut.