On Niko Kovac's return to his former club, Bayern took the lead midway through the first half when Ribery swept in at the culmination of a slick counter-attack.
Bayern could have been ahead earlier on, had Kevin Trapp not pulled off a superb double save to deny Javi Martinez and Thomas Muller, who later hit the woodwork.
Robert Lewandowski should have wrapped up the win early in the second half, but his blushes were spared when Ribery scuffed in a second, with Rafinha's fortuitous effort adding further gloss to a victory that takes Bayern back to within six points of leaders Borussia Dortmund.
Danny da Costa and Luka Jovic both went close as Eintracht started well, but the hosts would have been behind had Trapp not followed up a fantastic save from Martinez's diving header by denying Muller on the rebound.
Trapp was beaten in the 35th minute, though. Muller played in Lewandowski, who squared a wonderful reverse pass into the centre of the box, with Ribery on hand to tuck home.
3 - @FranckRibery scores in 3 consecutive games in the Bundesliga for the first time since February 2015, after not scoring in his first 11 games in the Bundesliga in this season. Form. @FCBayernEN #SGEFCB— OptaFranz (@OptaFranz) December 22, 2018
Lewandowski almost turned provider again on the stroke of half-time, but Muller could only direct his header onto the bar.
Ribery should have had an assist when he teed up Lewandowski, though the striker's first touch let him down, and that profligacy might have proved costly had Jovic connected with Da Costa's cross.
But any lingering hopes of an Eintracht comeback were ended when Ribery's mishit clipped in off the woodwork, with Rafinha's cross-shot then getting the better of Trapp as Bayern cruised to a fifth successive Bundesliga win.
What does it mean? Kovac keeps the bragging rights
Kovac has now come out as the winning manager in the last three meetings between these sides, once as Eintracht boss in the DFB-Pokal final, and now twice during his time in charge of Bayern.
It also means that Bayern ensure that they stay within touching distance of Dortmund heading into the winter break, while Eintracht sit in fifth.
Ribery's scoring run rolls on
It took until December 8 for Ribery to open his account for the season, but the winger has not looked back since then, and has now netted four times in his last three Bundesliga appearances.
Haller, Jovic left isolated
After initially struggling to cope with Eintracht's high press, Bayern managed to stem the service into the hosts' forward pair, and neither Sebastien Haller or Jovic had much to go off.
Key Opta facts:
- Thomas Muller played his 50th competitive game for Bayern in 2018 – no other player made so many appearances for Bayern in all competitions this year.
- Franck Ribery scored in his third consecutive Bundesliga game – the last time he achieved this was in February 2015.
- Robert Lewandowski had a hand in a goal for Bayern for the 50th time in all competitions in 2018 – tying his personal record for a single calendar year (50 goals scored/assisted in 2017).
- Ribery (35 years, 259 days) became the oldest Bayern player ever to score at least
two goals in a single Bundesliga game.
- Niko Kovac has won five consecutive Bundesliga games for the first time as a coach – he also did not achieved this in his career as a Bayern player.
With the Bundesliga taking a winter break, Bayern face Fortuna Dusseldorf in a friendly before restarting league duties away at Hoffenheim on January 18.