Dutch 21-year-old Verstappen, who had started from pole position for the first time, was runner-up after being reeled in and overtaken four laps from the end with his tyres worn to the limit.
"That was definitely a tall order but I'm grateful we did it," said Hamilton of a strategy that left him with a gap of almost 20 seconds to close in the final 22 laps.
"Man, that feels so good, guys."
Ferrari's Sebastian Vettel finished third and more than a minute behind - an eternity in Formula One terms - as the top two waged their own private battle in the heat of the Hungaroring.
Hamilton, a five-times world champion who also triumphed in Hungary last year, now has 250 points, with team mate Valtteri Bottas on 188 and Verstappen closing in fast on 181.
The win was Hamilton's eighth in 12 races this season and the 81st of his career - 10 short of seven-times champion Michael Schumacher's all-time record.
Bottas, starting on the front row, finished a forlorn eighth after a fight back from last place following an early stop to change a front wing damaged as he and Hamilton made light contact on the opening lap.
Hamilton, using fresher and faster medium tyres on a switched two-stop strategy after starting third, lapped all but the three cars behind him in a complete turnaround from his poor performance in Germany the previous weekend.
"It was difficult to get by, the defence was great, and I didn't know if I could catch that 19-second gap," said the Briton of a battle with Verstappen that ebbed and flowed throughout the race.
"I just kept my head down, and each lap was like a qualifying lap. I know if (late Mercedes non-executive chairman and triple champion) Niki (Lauda) was here today, he would take his hat off."
Verstappen, who made a late pitstop after being passed and made sure of an extra point for the fastest lap, crossed the line 17.7 seconds behind Hamilton.
"They rolled the dice and it worked for them, unfortunately. But you drove your heart out," team boss Christian Horner told the youngster.
"I think Lewis today was on fire as well," Verstappen told reporters.
"We were just not fast enough but I tried everything I could on those hard tyres to stay alive. Unfortunately it wasn't enough but, to finish second with the fastest lap, it was a good weekend overall."
Any threat from Bottas was snuffed out on the opening lap when first Hamilton muscled through and then Ferrari's Charles Leclerc, who finished fourth, made contact as he passed the Finn.
