Burak Yilmaz had given Turkey the lead eight minutes earlier, but Bode ensured honours were shared as both sides looked for a win to improve their play-off hopes.
Hungary had the first chance in the sixth minute, but Adam Szalai hit his shot straight at Turkey goalkeeper Onur Kivrak.
Turkey forced a succession of corners and then Alper Potuk shot wide for the home side, who lost the reverse fixture in October 3-1.
With nearly 20 minutes gone Szalai tried another shot from distance, which was again off target, as was Turkey's Potuk when he shot from long range.
But just three minutes later Hungary twice went close to taking the lead, only to be denied by Turkey keeper Kivrak.
Balazs Dzsudzsak's free-kick was headed by Szalai but the effort was parried by Kivrak, who also saved Tamas Kadar's follow-up shot.
Nuri Sahin shot over the bar and then Selcuk Inan saw his effort blocked by Hungary defender Jozsef Varga.
Hungary keeper Gabor Kiraly parried a shot from Arda Turin before Yilmaz shot wide.
Turkey should have taken the lead, but Potuk missed the target with a shot from close range and then Kiraly tipped over a 25-yard shot from Nuri Sahin.
Yilmaz shot over the bar and Hungary's Tamas Hajnal fired wide as both sides started the second half well.
Yilmaz finally made the breakthrough on the hour when he headed in Arda Turan's cross from close range.
Hungary were level with 20 minutes to go when substitute Bode shot home from close range
The visitors nearly took the lead when Vladimir Koman's shot was blocked, but neither side could break the deadlock in a frenetic finish.