Kvyat, meanwhile, set an identical lap time to Ferrari’s Kimi Raikkonen in third. Sebastian Vettel was half a second back on team mate Raikkonen in fourth, with no other drivers getting within a second of Verstappen’s benchmark.
The 90-minute session began with teams using intermediate tyres on a damp-but-drying track. By the end - on slick rubber - the cars were reaching speeds of well over 350 km/h in the rarefied air of Mexico City.
The high altitude presented a few cooling issues, with the most dramatic-looking affecting Rosberg, whose rear brakes caught alight. The fresh track surface also made for plenty of spinners.
Watched by a sizeable crowd of enthusiastic spectators, Pastor Maldonado led the field out at the start, bang on 1000 hours, amid much wild waving of flags, heralding the first official F1 running in Mexico since 1992.
The green track was very slippery in the early going, and everybody bar Williams’ Valtteri Bottas ventured out on intermediate tyres. The Finn opted for full wets.
“The grip is unbelievable, really, really low and not just in the corners,” team mate Felipe Massa reported.
After 12 minutes Raikkonen was the first to set a time, lapping his Ferrari in 1m 37.169s. Verstappen soon beat that with 1m 35.283s, though he appeared to be outside track limits in several slippery corners. Vettel slotted into second, on 1m 35.845s.
As conditions continued to improve, Kvyat set the pace after 16 minutes with 1m 34.041s for Red Bull, before Carlos Sainz went to the top with 1m 33.095s for Toro Rosso. Kvyat improved to 1m 33.719s to stay second until Rosberg posted 1m 33.174s.
After 20 minutes Williams sent both their drivers out on the medium slick Pirellis and the result was a fastest time of 1m 32.489s for Bottas. Ferrari followed suit, and as Bottas improved to 1m 31.868s, Vettel went sixth on 1m 34.032s. Cue wholesale changes to the slick rubber, even though the track was still damp enough to provoke slides and wheelspin. Vettel improved to second, on 1m 32.161s, just after Bottas went faster still on 1m 30.678s and Massa went third on 1m 32.580s.
But the Mercedes were now out, and after 28 minutes Hamilton’s first flier put him second on 1m 31.160s ahead of Sergio Perez, the crowd’s darling, on 1m 32.034s in the Force India. Hamilton then went purple in the first two sectors, but pitted at the end of that lap as he was just out of time on that set of tyres.
At the end of the first half hour the top five read: Bottas, Hamilton, Perez, Felipe Nasr and Vettel. Yet to record times were Jolyon Palmer in Romain Grosjean’s Lotus, and Nico Hulkenberg’s Force India. The latter had problems with its brake temperatures.
After a short lull Ricciardo came out and moved quickly from 12th to third to first, with 1m 30.577s, as the track improved further. Sainz jumped to third with 1m 30.735s, Verstappen to fourth with 1m 30.782s, before Ricciardo did 1m 29.502s to stay first, and Maldonado banged in 1m 30.271s to go second before improving to 1m 29.824s.
Out came the Mercedes again, with just over 40 minutes remaining. By then Hamilton had dropped to eighth, and Rosberg was down in 18th - but not for long. Hamilton’s first lap put him third on 1m 30.126s, just as Fernando Alonso jumped to second with 1m 30.072s for McLaren. Then Palmer went quickest, on 1m 29.163s, before Rosberg topped the times with 1m 28.399s, at one stage resorting to the run-off area to pass Hamilton. It was one of those moments when every lap brought faster pace as the track continued to get better.
With 35 minutes left, Verstappen moved to second on 1m 28.744s, Massa to third on 1m 29.026s.
Then Bottas popped up at the top of the timesheets again, with 1m 27.708s. Hamilton climbed to third with 1m 27.773s, before Rosberg moved back to the top spot once more with 1m 27.226s.
Rosberg’s efforts to go quicker resulted in an off-track moment in Turn 12, which leads into the stadium section, and he was very lucky to get it all stopped before his Mercedes touched the wall. He swept back to the pits with his rear brakes afire, with Mercedes urgently instructing Hamilton to cool his own brakes, which were 'on the limit'.
Meanwhile, Bottas was second on 1m 27.303s with Verstappen flying in third on 1m 27.381s. To the delight of the crowd, Perez was fifth on 1m 27.581s.
With 27 minutes to go Kvyat again put Red Bull on top with 1m 27.181s, then trimmed that to 1m 26.663s and then 1m 26.295s. Ricciardo joined him, with 1m 27.185s. Curiously, Alonso went out for a few laps on intermediates, after damaging one of his slicks over the numerous kerbs.
With 15 minutes remaining Mercedes sent Hamilton back out to check out the revised cooling on his F1 W06 Hybrid. As he set about his out-lap he was demoted to ninth as Sainz vaulted him for sixth, while Bottas spun harmlessly in Turn 3 chasing the Mercedes.
The stage was set for more improvements. Verstappen jumped to second with 1m 26.602s; Kimi Raikkonen to third with 1m 26.877s; and Vettel to fourth with 1m 27.127s. Hamilton was now 11th, however, and showing little sign of imminent improvement.
Then Verstappen went fastest with 1m 25.990s, though he’d missed a chunk of Turn 11 in doing so. Mercedes got Rosberg out right at the end, and he improved to sixth, hitting 342 km/h on the straight in the process.
Thus, after a non-stop action session, the top 10 was Verstappen, Kvyat, Raikkonen, Vettel, Ricciardo, Rosberg, Bottas, Sainz, Perez and Massa. Hamilton stayed 11th, while further back Button was 19th after stopping early to have his engine changed, and Will Stevens sat in the Marussia garage with technical problems for much of the session and only got going in the final 15 minutes.