Goals by Venezuela striker Juan Arango, attacking midfielder Marco Reus and defender Tony Jantschke gave Gladbach their first Bundesliga win in six games, while Cologne suffered their fifth defeat in six.
With only three matches left, Gladbach remain fourth, seven points clear of fifth-placed VfB Stuttgart, and they will be able to book a place in the Champions League qualifying rounds if they hold their place.
They are just one point behind Schalke, who in third place in the Bundesliga automatically get a place in the Champions League.
"Everything is still possible, but we will continue to think from game to game," said Gladbach coach Lucien Favre.
"It was a deserved victory. We won 3-0, which is what we wanted to do, but it was not easy."
Defeat leaves Cologne 16th and heading for a relegation play-off at the end of the season against the team that finish third in Germany's second division.
Cologne replaced Norwegian coach Stale Solbakken with Frank Schäfer on Thursday, but the 48-year-old has his work cut out, with Cologne at home to resurgent Stuttgart - unbeaten in their last nine games - next weekend.
Both Gladbach and Cologne are set to lose their star players at the end of the season, with Reus set to leave Gladach for defending champions Borussia Dortmund and Cologne's Germany star Lukas Podolski tipped to join Arsenal.
Dortmund can cement their place as German league champions on Saturday if they beat Gladbach at home, after a 2-1 victory at Schalke left them eight points clear at the top.
"It feels good, but no, nothing has been decided yet," insisted Dortmund coach Jürgen Klopp, whose team set a new Bundesliga record of 25 games without defeat in a single season.