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.
Delhi were 83 for two at half-way mark after 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 on, Chris Morris (16 off 8 balls) and Anderson took the onus to get Delhi to a fighting total, hitting some lusty hits towards the end of innings.
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. The hosts collected 68 runs from the last five overs, with the final one yielding 19.
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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.
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.
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