Following a mid-race red flag caused by Kevin Magnussen crashing his Haas, Hamilton took to the restart from third spot with the medium tyre as Charles Leclerc and Verstappen ahead of him opted for the hards.
Although he failed to make the most of the early tyre advantage to leapfrog Leclerc away from the grid, Hamilton eventually managed to get past the Ferrari a few laps later as he then set sights on Verstappen ahead.
However, the world champion had a clear and obvious pace advantage and slowly pulled clear, as Hamilton then knuckled to keep his tyres alive.
Although the pursuing Ferrari team was confident that the mediums would go off to help its drivers recover, Hamilton managed to keep them alive and even bagged the fastest lap on the final lap.
Speaking to his former team-mate and 2009 world champion Jenson Button in the post-race television interviews, Hamilton admitted it had not been easy.
"I wasn't thinking it was going to last," he said about that final stint on the mediums. "But I did some real sweet finesse. I was trying to drive like you..."
Hamilton's second-place finish was a welcome boost after he got disqualified from an identical result at last weekend's United States Grand Prix.
And he said he was delighted with the way Mercedes had knuckled down after the Austin frustration to deliver another strong result.
"I feel fresh," he said. "We weren't having to push crazy hard out there. But this is a great result considering we started sixth.
"I'm really proud of the team. This is a really difficult couple of weeks and to bounce back from a difficult weekend last weekend with the end result, this is really well done."
Race winner Verstappen said he was pretty untroubled on his way to victory, as he secured a record-breaking 16th win in an F1 season.
"I mean, to be honest, we are of course experiencing I think an incredible season," he said.
"Today as well, of course, we had to start P3. But I think the pace of the car was very, very good. We tried to do a different strategy to everyone else and unfortunately, with a red flag, we couldn't really show it basically.
"But nevertheless, on those hard tyres at the end, I think we were very, very strong."
