Billings (55) struck his second IPL fifty before Anderson (39 not out) provided late fireworks to lead the hosts to a respectable 188 for six.
The hosts collected 68 runs from the last five overs and it tilted the game in their favour after mid-overs struggle.
Daredevils, who have one of most potent attacks in the league, then restricted Punjab to 137 for nine to hand the visitors their second straight defeat.
The home spinners as well as pacers dominated the Punjab batsmen, never giving them free runs.
It was spinner Shahbaz Nadeem who rattled the Punjab top-order by scalping Manan Vohra (3) and Wriddhiman Saha (7) in successive overs while Chris Morris (3/23) saw the back of Hashim Amla (19).
Adding to the wickets, skipper Zaheer Khan and Australian quick Pat Cummins (2/23) bowled tight overs to choke Punjab.
Punjab's 50 came as late as in the ninth over while Eoin Morgan (22) departed at the half-way mark, caught by Karun Nair off Cummins.
Tottering at 64 for four in 10 overs, the visitors needed 125 runs from the last 10. With David Miller and skipper Glenn Maxwell around it was not impossible but Daredevils had too good an attack.
Leg-spinner Amit Mishra, who has been very consistent from the start of the league, dealt a severe blow to Punjab by seeing the back of Maxwell, who could not even open his account.
The last nail in the Punjab coffin was the wicket of Miller, who was trapped by Anderson in the 14th over.
Earlier, the hosts after a surprise decision to bat first, enjoyed their best start of the season through Billings and Sanju Samson (19) but intermittent wickets hurt them a bit.
They were 83 for two after 10 overs, following the opening pair's 53-run stand but from overs 10 to 15 the hosts managed only 37 runs and lost the set pair of Shreyas Iyer (22) and Billings.
From there, Morris and Anderson took the onus to get the side to a fighting total, hitting some lusty hits towards the end of innings.
Morris' cameo together with Anderson's calculated approach negated the good work done by Punjab spinners Axar Patel (1/33) and KC Cariappa (1/23).
Anderson, who hit three fours and as many shots over the ropes in his unbeaten knock off 22 balls, ended the innings with a six off Sandeep Sharma.
At the start, both Punjab pacers Sandeep and Mohit Sharma were largely disciplined but a marginal error was enough for Samson and Billings to steal boundaries.
Billings, who had been struggling for form so far, took some time to find his rhythm but gradually came into his zone. Samson carried on with his tremendous touch after hammering his maiden IPL ton against Pune and began with two fours off Sandeep.
Punjab today fielded five specialist bowlers and skipper Glenn Maxwell quickly shuffled them around. The Punjab pacemen though proved ineffective. Billings made Varun Aaron look pedestrian with a hat-trick of fours in the fifth over.
Delhi openers put on 49 after Powerplay overs as Samson ended the sixth over with a six off Mohit. Left-arm spinner Patel was the only bowler who got some respect from the home batsmen.
Leg-spinner Cariappa, who came in for Ishant Sharma, provided the breakthrough by scalping Samson, who miscued one to long-off. Aaron extended the poor run of Karun Nair (0) by having him caught by Wriddhiman Saha.
Billings found steady partner in Iyer but the Delhi youngster lost his wicket while attempting to accelerate off Mohit. Billings reached his fifty but could not make the most of a good start, becoming a victim of Patel.
The onus was on young Rishabh Pant (15) to take the side to a fighting total but he too fell soon. The wickets kept tumbling but Anderson succeeded in providing impetus to the innings.
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