The former Barcelona and AC Milan forward, leading scorer in the championship with eight goals, netted the equaliser and an 82nd minute winner as Flamengo fought back after Santos had raced to a 3-0 lead in 26 minutes.
Borges scored two goals for the hosts before Brazil's teenage forward Neymar weaved his way past four markers to add a brilliant third in Wednesday's match.
Flamengo pulled one back when Ronaldinho opened his account, taking advantage of a slip by Santos goalkeeper Rafael to score from close range, then Thiago Neves notched another for the visitors.
Brazil midfielder Elano missed a penalty for Santos before Deivid headed Flamengo's equaliser from a corner, all before the end of the first half.
The drama continued after the break as Neymar put Santos back in front, before Ronaldinho took command, equalising with a free-kick which went under the wall, then scoring the winner with an angled shot.
Flamengo are third on 24 points from 12 games, four behind leaders Corinthians.
