The winners of the game at Hampden Park in Glasgow will take on Austria or Wales for a place in November’s World Cup finals in Qatar.
Ukraine’s Football Association had asked FIFA to postpone the qualifier following Russia’s invasion of Ukraine.
“Following consultation with UEFA and the four participating member associations in Path A of the European qualifying play-offs, it was unanimously agreed in the spirit of solidarity to accept,” FIFA said in a statement.
“The match between Scotland and Ukraine … will now be postponed to the existing June window, and consequently, the match between the winners of Scotland v Ukraine and Wales v Austria will also be postponed.
“The match between Wales and Austria (March 24) will remain as scheduled … owing to the need to limit the disruption to the competitive matches already scheduled for the June window.”
Russia, who have been suspended from international soccer by UEFA and FIFA, were scheduled to host Poland in a qualifying playoff match on March 24.
FIFA added that Poland would receive a bye to the next round, and play either Sweden or the Czech Republic.
Three teams will qualify for the World Cup from the European playoffs.
The Russian Football Union has appealed against its suspension to the Court of Arbitration for Sport (CAS), which confirmed on Tuesday it had registered the case.
The draw for the World Cup finals is to be held in Doha on April 1.
Ukraine began playing as an independent nation in 1992 and last qualified for the World Cup finals in 2006, reaching the quarter-finals. The country made the last eight at Euro 2020 which was held last year.
Russia calls its actions in Ukraine a “special operation” designed not to occupy territory but to destroy its neighbour’s military capabilities and capture what it regards as dangerous nationalists.