Despite a disjointed performance in which they were reduced to 10 men with defender Jack Stephens' red card, they kept their third clean sheet from three to take their Group Nine record to two wins and a draw from their opening three fixtures ahead of Monday's visit of Switzerland.
The dropped points were their first in a qualifying fixture for two years.
England, playing without Eric Dier and Dele Alli who are now with Roy Hodgson's senior team and in the absence of the injured Jordan Ibe and Tyias Browning, struggled to play the fluid football manager Gareth Southgate encourages on a rather uneven pitch in Sarajevo.
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In a promising opening period, Lewis Baker threatened with a header at Bosnia goalkeeper Kenan Piric, and the hosts' Haris Hajradinovic went close when England's Jordan Pickford fumbled his shot.
Bosnia's Damir Sadikovic similarly sent a powerful effort over the crossbar soon after, before two of Southgate's most experienced players combined to go close again, this time with a header from Nathaniel Chalobah after captain James Ward-Prowse's accurate cross.
The hosts' patience and organisation presented a difficult challenge for Southgate's new-look team, and it soon began to appear that only a moment of true quality would create a clear opening.
Dominic Solanke, on loan at Vitesse Arnhem from Chelsea and making his England debut, caught the eye in that respect, and almost found space with a classy backheel in the final third.
That was demonstrated again approaching half-time when Calum Chambers headed over from another classy cross from Ward-Prowse, and when Chuba Akpom's heavy touch lost possession after Dominic Iorfa had created space.
In the second half, with the promising Ruben Loftus-Cheek disappointing, Armin Cerimagic's influence began to grow and England's ambitions waned.
Ward-Prowse curled wide from 20 yards in the 67th minute, and then Stephens, after a foul on Cerimagic, picked up the second of two yellow cards in quick succession to leave England to play with 10 men, against improving opposition, from the 72nd minute.
Another Cerimagic break, in the 82nd minute, ended with a cross into the area eluding his team-mates as England also became tentative amid the growing pressure.
With the last notable chance of the game, and in a repeat of an earlier move, Ward-Prowse again found Chambers in the area, but again the defender's header missed the target, this time going wide.
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