The Trotters, who boasted the worst scoring record in the division going into this game, nearly edged ahead after only four minutes when Derik Osede's cross picked out Darren Pratley, but the midfielder's header drifted just wide of Marco Silvestri's goal.
Bolton then suffered an early setback when former Blackpool frontman Gary Madine was forced off in the early stages, to be replaced by Ameobi.
However, it was Wanderers who continued to press and went close again as Pratley was again picked out in the box, only to send his header over the crossbar.
Midway through the half, Pratley then turned provider with a cross for Liam Feeney, but he also failed to hit the target as Bolton struggled to convert their early pressure into a lead.
The hosts finally edged in front after 32 minutes when a corner spun off the surface, which wrong-footed several Leeds defenders before allowing Ameobi to score from close range.