Victor Manuel Vucetich's men head into their final CONCACAF qualifier in fourth place following victory in the new manager's first game in charge against Panama on Friday.
Mexico are now in pole position for a spot in the intercontinental play-off with New Zealand in November, but appear unlikely to overhaul third-placed Honduras and seal automatic qualification.
Vucetich's side would need three points against Costa Rica, a victory for bottom-placed Jamaica over Honduras and a two-goal swing in their favour to finish third.
The visitors should take confidence from their success over Panama, but face a tough opponent in Costa Rica, who suffered only their second defeat of the final round of qualification in Honduras three days ago.
Jorge Luis Pinto's Costa Rica - who have already ensured their place at next year's finals - have also not lost at home since September 2012.
It was Mexico who triumphed on that day as they ran out 2-0 victors in the second round of qualification, but the hosts will be eager to exact revenge against a team they have not beaten since 2001.
Defeat for Mexico could allow Panama - who are three points behind in fifth ahead of their game with pool winners United States - to move into the play-off spot.
The onus will be on the likes of Javier Hernandez, who has scored five times in qualifying, and Raul Jimenez - scorer of the late winner against Panama - to help them avoid that scenario.
A draw will be enough for Mexico to wrap up their place in the play-off, but Vucetich's men will be determined to record successive victories for the first time since July in San Jose.
