Bradley's side were on course for a first win since the opening day of the league season.
Swansea manager Bob Bradley was denied his first Premier League victory in agonising fashion as Everton's Seamus Coleman hit a last-minute equaliser to rescue a 1-1 draw on Saturday.
Bradley's side were on course for a first win since the opening day of the league season when Gylfi Sigurdsson's penalty put them ahead at Goodison Park.
But Coleman netted in the 89th minute for Ronald Koeman's team, leaving former United States coach Bradley with just two points in his five matches since his surprise appointment as successor to the sacked Francesco Guidolin.
Adding insult to injury for Bradley, Swansea dropped to the bottom of the table following Sunderland's victory over Hull.
Bradley made eight changes from the side which lost to Manchester United last time out, including dropping South Korean Ki Sung-Yueng, who suggested while on international duty the club was in "crisis".
Sigurdsson looked dangerous from set-pieces and open play, having one free-kick deflected over by the wall, with Jordi Amat heading wide from the resulting corner.
Sigurdsson's value to Swansea cannot be underestimated -- since joining the Welsh side he has scored or assisted 37 of their 99 Premier League goals.
He often troubled Everton centre-backs Phil Jagielka and Swansea old boy Ashley Williams when he dropped deep to gain possession.
Jagielka was caught out five minutes before half-time to cap a troubled week that saw him photographed with Wayne Rooney during the England captain's now-infamous drinking spree.
Sigurdsson latched on to Modou Barrow's flick-on to burst into the area and, after he skipped past Jagielka, the veteran centre-back's clumsy challenge conceded a penalty that the Iceland midfielder coolly converted.
Romelu Lukaku, who had scored seven times in his previous nine league appearances, failed to react quick enough to Yannick Bolasie's cross, which he allowed to hit him and bounce over an open goal from three yards.
Bolasie was also not free from blame after shooting too close to Lukasz Fabianski when put clean through by Lukaku.
Swansea, sensing a long-awaited and much-needed three points, dropped deeper and deeper in the second half, throwing everything at the ball in a scrappy but effective rearguard action.
Just when they appeared to have held out, Amat's weak header dropped to Coleman, who nodded a looping effort back past Fabianski's reach.