Rudi Garcia's Roma, who were missing injured attackers Francesco Totti and Gervinho and had right back Maicon sent off for a second bookable offence in the 66th minute, stay top with 27 points after a gutsy display at the Stadio Friuli.
Napoli remain second, level with Juventus on 22 points, after two controversial Gonzalo Higuain penalties helped Rafael Benitez's side beat 10-man Torino 2-0, while an Arturo Vidal penalty and a fine Carlos Tevez goal gave champions Juventus 1 2-0 win over Genoa.
Bradley's winning strike was created by Kevin Strootman, who held the ball up well before laying off a perfect pass for the American midfielder.
"We're taking it one game at a time. We came here today looking to play well and get three points. We gave everything and if you ask me these are three deserved points," Bradley said in a television interview after the match.
"This is a team with great desire and determination. Last year it didn't go as well as it could have, and this year it's going a bit better. Now we're concentrated on Chievo and we'll give it our all."
Roma's victory equals the nine consecutive wins Juventus recorded at the start of their title-winning 2005-06 season.
They were later stripped of that title in the "Calciopoli" match-fixing scandal.
Napoli were awarded their first penalty after Dries Mertens looked to have gone down easily under Nicola Bellomo's challenge after 14 minutes, while a harsh handball against Kamil Glik earned them the second just after the 30 minute mark.
Glik accidentally blocked Federico Fernandez's shot from point blank range with his arm.
Referee Andrea De Marco initially waved away the home side's penalty claims but with Torino breaking away and bearing down on goal, he then pulled play back and pointed to the spot, infuriating Torino boss Giampiero Ventura.
Torino later had Migjen Basha sent off for a professional foul on Goran Pandev four minutes from time.
"The first one I don't know whether he touched him, and the second wasn't a penalty but it doesn't change anything because we were never in the game," Ventura said afterwards.
There was also controversy in the manner Juventus took the lead midway through the first half when Kwadwo Asamoah appeared to be brought down outside the area by Davide Biondini for Vidal's penalty.
There was no doubt about their second however, with Asamoah feeding Tevez from out wide, allowing the Argentine attacker to brilliantly slot the ball home with his left foot.
AC Milan 10th with 11 points after Marco Parolo's last-gasp free kick consigned Massimiliano Allegri's side to a 3-2 defeat at Parma.
Two goals in two minutes from Alessandro Matri, who had replaced Mario Balotelli in the 52nd minute, and Matias Silvestre had brought Milan back from two goals down against their hosts, who led at halftime through goals from Parolo and Antonio Cassano, against his former club.
"The first half we created little and they had three counter attacks and scored twice," said Allegri.
"I'm annoyed because the team played well in the second half...but after having put the effort in and equalising instead of holding on to the 2-2 we conceded from the free-kick. Parolo did well but we made mistakes."
A Juan Cuadrado brace either side of halftime helped Fiorentina go fifth after coming back from a goal down to win 2-1 at Chievo, who drop to bottom in Serie A after Bologna beat Livorno 1-0 and Sassuolo drew 0-0 at Catania.