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After losing David Warner to the first ball of the match, Capitals posted 159-7 with Mitchell Marsh scoring 63 off 48 and Sarfaraz Khan 32 off 16.
Liam Livingstone and Arshdeep Singh took three wickets apiece.
In reply, Punjab Kings were kept to 142-9 with Jitesh Sharma making 44 off 34, Jonny Bairstow 28 off 15 and Rahul Chahar 25*.
Bowling for Delhi Capitals, Shardul Thakur took 4-36, while Axar Patel and Kuldeep Yadav each had 2-14.
IPL action continues on Tuesday with Mumbai Indians playing Sunrisers Hyderabad from 10:00h.
