Atletico have been disappointing in LaLiga and again looked susceptible at the back in the first half, but their quality in the final third ensured they had too much for Leonardo Jardim's men.
A couple of mistakes in quick succession ruined Atletico's positive start as Samuel Grandsir netted on his Champions League debut, but the superior visitors deservedly went into the break ahead thanks to Costa and Gimenez.
That lead never looked in danger after the restart, as Atletico enjoyed complete control of the midfield and saw out the win.
Although Atletico started the brighter, Monaco punished two errors in the 18th minute as Radamel Falcao robbed Saul Niguez in the area and saw Angel Correa knock his cutback on to Grandsir, who prodded past Jan Oblak.
Costa made amends for an early miss by levelling just after the half-hour mark, finishing beyond Diego Benaglio with aplomb after Antoine Griezmann's cushioned first-time pass released him.
Atletico completed the turnaround on the stroke of half-time – Gimenez powering a header past Benaglio from Koke's corner.
The away side sat back more after the break, offering encouragement to Monaco, but a lack of sharpness in attack saw the hosts fail to capitalise.
Substitute Jordi Mboula looked lively after replacing Nacer Chadli, though the best he could muster was a long-range effort which sailed over and Atletico easily cruised to the final whistle.
What it means: Atletico mean business
Diego Simeone's side failed to get out of the group stage last season and, although that ultimately allowed them to win the Europa League, the coach will not want a repeat. They have taken a step in the right direction with a professional away win.
Pat on the back: Griezmann talks with his feet
Griezmann's recent public targeting of the Ballon d'Or attracted plenty of criticism, particularly given his underwhelming start to LaLiga's new season. But in Monaco he was a little more like his usual self, slicing the hosts open several times and linking brilliantly with Costa for his equaliser.
Boot up the backside: Ineffective Chadli
For a side playing at home, Monaco were particularly disappointing in attack. Chadli's presence on the right wing did little to help them, as the game passed the anonymous Belgian by before he was withdrawn for Mboula in the 58th minute.
Key Opta facts:
3 - Atletico de Madrid top-scorers in Champions League:— OptaJose (@OptaJose) September 18, 2018
1 - Antoine Griezmann: 17
2 - DIEGO COSTA: 9
3 - Saúl Ñíguez: 8
- Atletico had 58 per cent of possession against Monaco, their highest in a game this season in all competitions.
- Between Champions League and Europa League, Griezmann has been involved in seven of Atletico's last nine goals (five goals, two assists).
- Atletico players have made more errors leading to goals this season in all competitions (four) than the whole of the last campaign (two).
- Oblak has only kept a clean sheet in one of his last seven games for Atletico in all competitions (1-0 versus Rayo Vallecano), conceding nine goals in total.
Monaco host Nimes on Friday hoping to pick up their first win since the opening weekend of the Ligue 1 season, while Atletico – who also have just one league win so far this term – face a local derby at Getafe on Saturday.