The Royal's pacers -- Rahman (2/22), Chetan Sakariya (2/33), and Kartik Tyagi (1/40) -- shared five wickets while spin all-rounder Rahul Tewatia (1/17) dismissed one batsman.
Tyagi, the star of the match against Punjab Kings, drew first blood with his very first delivery in the fourth over, cleaning up Orange Cap holder Shikhar Dhawan (8).
The Delhi opener looked to play a good length delivery but the ball rolled back to hit the timber.
Prithvi Shaw (10) soon followed his opening partner back to the dugout, slicing the ball off Sakariya to Liam Livingstone at mid-off.
At the end of the powerplay, the Capitals were precariously placed at 36 runs for the loss of two wickets.
With spin introduced in the seventh over Shreyas Iyer (43 off 32 balls) and skipper Rishabh Pant (24 off 24 balls) started to repair the damage caused by the loss of the openers.
The duo, who stitched a 62-run partnership, kept the scoreboard ticking, while also hitting the occasional boundaries at the huge Sheikh Zayed Stadium.
The stylish Southpaw hit the first maximum of the match over long-off in the 10th over off left-arm wrist spinner Tabraiz Shamsi (0/34).
As the two Delhi batters found their footing, Rahman was brought back into the attack and the move immediately paid dividends as Pant, looking to hook the ball away, ended up getting an under the edge.
Samson then showed his skills with gloves to stump Iyer, who failed to stretch his back foot inside the crease as the wicket continued to tumble at regular intervals.
Shimron Hetmyer, who was striking the ball well, then upped the ante, hitting three boundaries off Tyagi to plunder 16 runs in the 16th over.
The big-hitting West Indian, who hit five fours, scored a brisk 28 off 16 balls.