Barcelona forward Duggan prodded home Melissa Lawley’s cross as England shrugged off Alex Greenwood’s red card to prevail in Zenica.
Jodie Taylor’s added-time penalty wrapped up a comfortable, albeit often frustrating, digital design afternoon in front of goal on the road.
Toni Duggan scored England´s opener (John Walton/PA)
Bosnia had Amira Spahic sent off too in a staccato encounter, with Phil Neville’s England faced with ultra-defensive tactics for the second match running.
The Lionesses had laboured to a goalless draw against Wales in Southampton on Friday night, unable to pull their rigid opponents apart at St Mary’s Stadium.
But despite being confronted with another defensive-minded 5-4-1 formation, this time on their travels, Neville’s side finally found a way to hit the net.
A frustrating first-half left England empty-handed, just as they had been against Wales on Friday.
Boss Neville had predicted Bosnia would set out two defensive lines of five players in a bid to thwart England, the world’s second-ranked outfit.
And so it proved, as the hosts attempted to follow Wales’ lead in keeping England off the scoresheet.
The visitors hogged the ball in that first-half, but also failed to make that dominance count, again struggling to create clear-cut scoring chances.
Greenwood’s red card acted as something of a catalyst for Neville’s team however, the Notts County defender handed her marching orders for two yellow cards.
Booked for dissent before the break, Greenwood was then dismissed for a second yellow card, for diving.
Despite consolation from Neville as she trudged off, Greenwood knew she had jeopardised England’s chances of claiming top spot in World Cup qualifying Group A.
But just when the visitors started to sweat on the source of that elusive goal, up popped Duggan, poking out a foot despite tight marking to stab home Lawley’s cross.
Bosnia were quickly reduced to 10 too, with Spahic sent off, again for two bookings.
England continued to dominate, and when Beth Mead won a late spot-kick, Taylor stepped up and duly fired home her 17th international goal.