Verstappen topped first practice with a time of 1:44.410, with team mate Daniel Ricciardo finishing four tenths behind him to ensure a Red Bull one-two, before the Dutch teenager capped an excellent day with an improved time of 1:43.362 in the second session.
The only disappointment for the pacesetter came when he appeared to lose control late on before spinning sideways into the advertising hoardings.
"It was very weird," Verstappen told Sky Sports. "I locked up and was still trying to make the corner, but I tried to play it safe and take the exit road.
"Suddenly I lost the rear and then I just had a four-wheel power slide into the barrier."
It was a scrappy day punctuated by several misjudgments on the scorching Baku circuit.
Force India's Sergio Perez had been flying through the city circuit and registered the fourth best time in the first session before crashing late on.
He escaped unscathed after colliding with the external wall at a narrow section at Turn Eight, and although his car lost its rear right wheel, Force India mechanics succeeded in repairing it for Perez to finish seventh in P2.
In the evening, Jolyon Palmer of Renault also crashed into the barriers after misjudging Turn Eight.
Championship leader Sebastian Vettel on the soft compound tyre registered times of 1:44.967 and 1.43.615, finishing third and fifth.
"I think it was very close today," he said. "I hope it's close tomorrow and hope we come out on top."
It was a mixed day for Mercedes duo Lewis Hamilton and Valtteri Bottas.
The Finn, who hit a Formula One record top speed of 373kmh last year, went into a spin at Turn 10 during the first practice, but recovered to finish second, a tenth behind Verstappen, in the evening's session.
Hamilton recorded the best time in each of the three practice sessions in Baku last year, before a crash in qualifying ruined his chances of success.
He looked uncomfortable on Friday, had a near miss with Ferrari's Kimi Raikkonen on Turn One and finished the second practice in tenth, 1.163 behind Verstappen.
"I think you need to get into a rhythm in Baku, do laps to gain confidence and have the tyres in the right window," said Mercedes boss Toto Wolff.
"We had a bit of a messy session, a lot of traffic, a lot of yellow flags and we didn't put him out at the right time."
McLaren-Honda duo Fernando Alonso and Stoffel Vandoorne were both slapped with a 15-place penalty and are likely to start Sunday's race at the back of the grid after picking up more engine penalties.
(Reporting by Ed Dove; Editing by Toby Davis and Andrew Heavens)
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