Chasing just 130 for victory, the visitors rode on Bairstow's blitz to cross the line with nine balls to spare and register their third straight win this edition after losing their first game against Kolkata Knight Riders.
Earlier stand-in skipper Bhuvneshwar Kumar (2-27), Mohammad Nabi (2-21) and Siddarth Kaul (2-35) restricted Delhi to 129-8 after choosing to field first.
"Really pleased with the start, just going to try and keep going," Bairstow said after receiving the man of the match award.
"We have three games in quick succession now and hopefully we keep playing the same brand of cricket. We have a great bunch of guys and we hope to keep going."
The English wicketkeeper-batsman, who hit 114 in his team's previous game against Royal Challengers Bangalore and was involved in a record 185-run opening stand with David Warner, put on 64 runs stand for the first wicket with the Aussie.
Bairstow smashed nine fours and a six before he was trapped lbw off Rahul Tewatia.
"Jonny is obviously in red-hot form coming on the back of a hundred. When you are confident like that, you don't think about the negative result," Delhi coach Ricky Ponting told reporters after his team's third loss in five matches.
"Running down the wicket, he is not even thinking that he is going to miss the ball, always thinking about that he is going to hit it in the middle for a six."
Warner looked out of sorts before falling to Kagiso Rabada for 10 off 18.
Delhi struck back with three more wickets but Yusuf Pathan (9*) and Nabi (17*) completed the chase.