Jonatan Johannsen hit the post for the visitors soon after, but he got his reward for some hard work, when his deflected shot made it 1-1.
Paul Bernard then ended his current good form, when he was sent off for a wreckless foul on Van Bronckhorst, putting the Dons down to 10 men.
Aberdeen reached half-time at 1-1, and were looking quite good despite being down to 10 men.
However Michael Mols struck for the Scottish Champions after 67 minutes to make it 2-1.
2 minutes later, Rangers score was extended to 3-1 when Amatostruck past helpless ex Scotland International Jim Leighton.
Johannson completed his hat-trick with two goals in as many minutes, striking with 12 and 10 minutes remaining.
Despite the final result, the Dons put up a brave fight, and the result possibly flattered the league leaders.