The 2018 World Cup runners-up took an early lead through a Luka Modric penalty, but the hosts hit back with Tamkin Khalilzada's brilliant 72nd-minute equaliser to claim their first point in Group E.
With the top two in each group to automatically qualify for next year's finals, Croatia are still top of the standings.
But they only lead third-placed Wales and fourth-placed Slovakia by four points, with both those teams having played a match less.
Hungary sit second in the table, one point behind Croatia, also with a game in hand, with the two teams to meet in Croatia on October 10.
"It wasn't a good match for us, especially the second half when we didn't take any chances," coach Zlatko Dalic told television channel HRT.
"We didn't score the second goal... And now we have to go on."
Azerbaijan, coached by Croatian Nikola Jurcevic, remain rooted to the foot of the table.
Dalic named a strong side to face the outfit ranked 109th in the world, including Real Madrid midfielder Modric, Bayern Munich winger Ivan Perisic and Liverpool centre-back Dejan Lovren.
But despite having 21 attempts at goal, Croatia could not break through again after Modric scored an 11th-minute spot-kick on his 34th birthday and paid the price.
They had thrashed Slovakia 4-0 away from home on Friday, when Azerbaijan narrowly lost 2-1 at Wales as Gareth Bale scored a late winner.