The visitors had fallen behind against the run of play in the 18th minute when winger Samuel Grandsir forced the ball home in a goalmouth scramble after good work from Radamel Falcao.
But Costa then ran onto a wonderful flick-on from Griezmann to fire a snap-shot low and hard past Monaco’s onrushing Swiss stopper Diego Benaglio to level the scores on the half-hour.
Jose Maria Gimenez — remembered for crying on the pitch when France beat Uruguay 2-0 in the World Cup quarter-finals in Russia — then headed home a corner in first-half stoppage-time to take the wind out of Monaco’s sails at a key moment.
“It’s always difficult to win the first match of the Champions League away from home,” said Atletico coach Diego Simeone.
“Griezmann is growing every game. He’s a decisive player. He did well in the right moments and his link-up with Costa was good.”
The pitch cut up worryingly and groundsmen scrambled at half-time to pound in sods of detached turf.
“The game was difficult, the pitch was not good for both teams, there was a danger of injury for the players,” said Simeone.
Former Monaco striker Thierry Henry and ex-coach Didier Deschamps were in the stands where just 12,000 fans showed up and amongst them a mere 80 Atletico supporters.
Atletico have had a stuttering start to their La Liga campaign, and with Simeone on the final match of a four-game touchline ban, the visitors were marshalled by the menacing bulk of his brooding assistant German Burgos prowling the line.
Monaco’s Polish defender Kamil Glik appeared to be relishing his role, keeping tabs on Costa and apart from the goal did a solid job on the Atletico target man.
Falcao never stopped searching for an equaliser against his old club, almost earning a penalty in the 66th minute and lashing a shot just wide moments later.
Glik saw a last-gasp header go just wide with the fans on their feet, but Atletico were otherwise untroubled in the latter stages.
France international forward Thomas Lemar was back at the Stade Louis II after his close-season switch to Madrid and came on late as Atletico sweated out the final minutes to clinch three precious away points.
For more news your way, download The Citizen’s app for iOS and Android.
Download our app and read this and other great stories on the move. Available for Android and iOS.