There is no nonstop path connecting the Potomac to the Miljacka River, which is precisely what makes booking business class from Washington Reagan to Sarajevo such an interesting exercise in routing strategy. Reagan National itself only handles a slice of the journey — you'll connect through a major gateway, most logically Munich or Frankfurt on Lufthansa, or Istanbul on Turkish Airlines — before the final descent into Sarajevo's compact, single-runway airport. Factually speaking, no US carrier flies this corridor directly, so Delta passengers will typically rely on a codeshare handoff to a Star Alliance or Lufthansa Group partner in Europe rather than a pure Delta metal experience. My own preference, after flying this corridor a dozen times, is the Lufthansa routing through Frankfurt, where the connection window is generous enough to shower in the business lounge before a short intra-European hop that often operates with recliner-style regional business seating rather than lie-flat, since the Frankfurt-Sarajevo leg is short-haul equipment.
The long-haul segment is where the real decision lies. Turkish Airlines via Istanbul gives you their widebody business cabin with the brand's signature lounge experience at IST — arguably the most entertaining airport lounge in the world, complete with a putting green and live cooking stations, worth arriving two hours early to enjoy. Lufthansa's transatlantic legs out of Dulles or similar US gateways increasingly feature the new Allegris business suites on select aircraft, though you'll want to confirm equipment before booking, as older 747 and A340 configurations still fly some rotations. Either way, budget real time in the connecting hub; Sarajevo's airport is small and efficient, but immigration during peak summer festival season can back up quickly.
January through March and again in October and November are consistently the softest months for fares on this corridor, as leisure demand into Bosnia collapses outside the summer and Sarajevo Film Festival crowds. A lesser-known trick: search Zagreb or Vienna as alternate final destinations when pricing Sarajevo itineraries, since carriers sometimes price the shorter regional add-on more aggressively than a direct SJJ fare bucket. Given how fare classes shift week to week on these connecting itineraries, setting up automated fare alerts is the smartest way to catch a premium-cabin deal before it disappears.