Hamilton ended the one-hour running at the Autodromo Hermanos Rodriguez just 0.076 seconds behind team-mate Valtteri Bottas as Mercedes recorded an encouraging one-two finish.
The high-altitude venue in Mexico City which sits a breathless 2,285 metres above sea level, had been expected to suit Verstappen’s Red Bull machinery.
But the Dutchman finished 0.123 sec adrift of Bottas to provide Hamilton with hope he can stop his rival from extending his 12-point title lead with just five rounds remaining.
However, the seven-time world champion faces a stewards’ investigation for failing to FIA race director Michael Masi’s notes.
In the early exchanges, Hamilton ran off at the opening right-hander, but instead of going round the bollard at Turn 3, he incorrectly rejoined the circuit at the following bend.
The stewards will delve into the incident ahead of Friday’s concluding action which takes place at 3pm local time.
Formula One is back in Mexico City for the first time in two years after last season’s event was cancelled amid the coronavirus outbreak.
And the majority of the support here is for home favourite Sergio Perez.
But the Red Bull driver spent half the session on the sidelines after he lost control of his car through the left-hander Turn 16 and slid backwards into the barriers.
After a 26-minute repair job, Perez rejoined the action to finish fourth, 0.269 secs off the pace.
Ferrari’s Charles Leclerc also suffered rear wing damage when he endured a similar accident to Perez. The Monegasque driver ended the running in eighth.
Elsewhere, Pierre Gasly finished fifth for AlphaTauri, one spot ahead of Ferrari driver Carlos Sainz with double world champion Fernando Alonso seventh in his Alpine.
British driver Lando Norris was only 15th for McLaren, the best part of two seconds back, with George Russell 17th of the 20 runners.