The Mercedes superstar followed up his Sao Paulo success with another triumph, although Verstappen limited the damage despite a five-place grid penalty due to a yellow-flag infringement in qualifying.
The championship leader recovered to finish second and also took the extra point for the fastest lap, assured of beating Hamilton to that feat due to a late virtual safety car following a series of punctures.
The two title rivals were joined on the podium by Fernando Alonso, the two-time champion's first top-three finish since 2014.
Both he and pole man Hamilton enjoyed strong starts, although Verstappen wasted little time in wiping out his penalty.
Starting from seventh, the Dutchman gained three places from the start but was then shut out by Alonso, who passed Pierre Gasly into second.
Verstappen stayed on Gasly's tail and eased past on lap four before quickly hunting down Alonso, yet Hamilton had already disappeared into the distance.
Hamilton continued to stretch his advantage out towards 10 seconds before Verstappen pitted and the Mercedes man immediately responded, following him in to maintain the lead.
That gap closed very slightly as Hamilton worked his way through the backmarkers, but the drama was limited at the front until Valtteri Bottas, running in third, suffered a puncture as he attempted a one-stop strategy.
That was the first of several such issues, meaning Verstappen took no risks and pitted a second time soon after, with Hamilton again heading in on the next lap.
Sergio Perez questioned Red Bull's call to bring him in twice as he pursued a podium, and it was a move that ultimately did not pay off as he failed to reel in Alonso due to the virtual safety car.
BOTTAS BLOW ADVISES STRATEGY
Mercedes expected a two-stop from Verstappen, so promised Bottas there would be "opportunities" if he went longer and took care of his tyres. Unfortunately, that strategy failed.
Bottas, who qualified in third, was penalised to sixth and then fell to 11th, had recovered to third place on lap 36 when he sustained a puncture and was forced to limp back to the pit, ruled out of contention entirely as he re-emerged in 14th. He later retired, just as Williams pair George Russell and Nicholas Latifi each also suffered punctures to the same front-left tyres.
Perez might have been less than impressed, but the single stops proved a huge gamble.
ADD LOSAIL TO LEWIS' LIST
Hamilton already held the record for poles (30 of 34) and wins (29 of 34) at the most different circuits and added to both tallies at the first-ever Qatar Grand Prix.
Alonso also enjoyed his Losail bow, collecting points at the 34th of 34 tracks in real style.
1. Lewis Hamilton (Mercedes)
2. Max Verstappen (Red Bull) +25.743s
3. Fernando Alonso (Alpine) +59.457s
4. Sergio Perez (Red Bull) +62.306s
5. Esteban Ocon (Alpine) +80.570s
6. Lance Stroll (Aston Martin) +81.274s
7. Carlos Sainz (Ferrari) +81.911s
8. Charles Leclerc (Ferrari) +83.126s
9. Lando Norris (McLaren) +1 LAP
10. Sebastian Vettel (Aston Martin) +1 LAP
1. Max Verstappen (Red Bull) 351.5
2. Lewis Hamilton (Mercedes) 343.5
3. Valtteri Bottas (Mercedes) 203
4. Sergio Perez (Red Bull) 190
5. Lando Norris (McLaren) 153
1. Mercedes 546.5
2. Red Bull 541.5
3. Ferrari 297.5
4. McLaren 258
5. Alpine 137
After a triple-header that saw the momentum swing back in Hamilton's favour, there are two weeks before the next race, the Saudi Arabian Grand Prix.