Vaclav Cerny scored the only goal at Ibrox in the opening minute of the second half but the hosts missed several other chances, hitting the woodwork on a remarkable four occasions.
Rangers are a point above Aberdeen but still eight points behind champions Celtic, who are away to Dundee United on Sunday.
"We had good chances but you need the second goal to kill the belief of the opponent, that's what we missed today," Rangers manager Philippe Clement told the BBC.
"We had enough opportunities for that.
"The important thing is the three points and it was well deserved. Defensively we did a good job.
"We always want to give our fans more than 1-0. We want to score three, four. The crossbar didn't help today."
Aberdeen made a storming start to the season, threatening to disrupt the Celtic and Rangers duopoly.
But the Dons' winless run was extended to six games on Saturday as they gave up the lead in a 3-1 loss at home to Hibernian.
Topi Keskinen put Aberdeen ahead in the 14th minute at Pittodrie but Elie Youan quickly equalised and Nicky Cadden and Martin Boyle secured victory for Edinburgh club Hibs with goals either side of half-time.
In Saturday's other game, Oisin Smyth scored a last-gasp winner as 10-man St Mirren snatched a dramatic 2-1 win at Ross County.