Triggers controlled by your opponent will resolve first, so you cannot use Obzedat's ability to exile it before the Archfiend's ability resolves. Rule 603.3b says
If multiple abilities have triggered since the last time a player received priority, each player, in APNAP order, puts triggered abilities he or she controls on the stack in any order he or she chooses. (See rule 101.4.) Then the game once again checks for and resolves state-based actions until none are performed, then abilities that triggered during this process go on the stack. This process repeats until no new state-based actions are performed and no abilities trigger. Then the appropriate player gets priority.
"APNAP" here means Active Player, Non-Active Player. When the abilities trigger, first the active player (the player taking the turn) puts abilities they control on the stack, then the other player does. This means that your Obzedat's ability goes on the stack, followed by the Archfiend's ability, so the Archfiend's ability resolves before you have a chance to exile Obzedat.