Dominic Solanke put Bournemouth ahead just seconds after half-time and his goal looked it might be enough to take the visitors to the summit in what was manager Scott Parker's first match at Craven Cottage since stepping down as manager of the London side in June.
But defender Adarabioyo headed in a cross from substitute Tom Cairney to ensure Fulham remained a point ahead of the Cherries.
Only the top two at the end of the regular season are guaranteed promotion to the lucrative Premier League, with this result leaving Bournemouth eight points clear of third-placed Queens Park Rangers.
This was a third successive draw for Marco Silva's Fulham, while the Cherries have won only one of their past six games following a fine start to the season.
A lively first half saw both sides create chances but, thanks to some fine defending and goalkeeping, the game was still scoreless at the interval.
That all changed when Bournemouth scored from the kick-off of the second half.
Following a neat exchange of passes, Philip Billing's ball over the top saw Solanke outpace Denis Odoi and lash a shot high past Marek Rodak for his 17th goal of the season.
Aleksandar Mitrovic twice went close for Fulham with a couple of headers until, six minutes from time, Adarabioyo powered in a header from seven yards out following a cross by Cairney.