Fast bowlers Mukesh Choudhary, Simrajeet Singh and Dwayne Bravo took two wickets each.
Chennai, who had a change of captaincy midway as Dhoni took over from Ravindra Jadeja, have only a slim chance of making the play-offs but improved on their net run-rate and moved to eighth in the 10-team table.
"You just want to enjoy the IPL. When two other teams are playing, you don't want to be under pressure," Dhoni said on his team's play-off hopes.
"You just have to think about what to do in the next game. If we make the playoffs, great. But even if we don't it's not the end of the world."
Delhi, who came into the game after a positive Covid test to one of their net bowler and with opening batter Prithvi Shaw in hospital with a fever, stay fifth as the Twenty20 tournament moves towards the business end.
Coach Ricky Ponting said his team will never make excuses and have earlier faced a similar situation when Delhi's New Zealand import Tim Seifert tested positive just hours before the start of a match last month.
"That's not an excuse for us. It's not the first time that we've had to deal with it on game day. In one of the earlier games against Punjab Kings we had to deal with the same thing and ended up winning that game really comfortably," Ponting told reporters.
"Although today wasn't ideal preparation, we need to play better cricket. To lose by 91 runs and put a huge dent in our net run-rate means we need to bounce back really strongly in our next game"
Conway set up the victory for Chennai with a 110-run stand with opening partner Ruturaj Gaikwad, who made 41, before Shivam Dube and Dhoni exploded.
Dube hit a 19-ball 32 in another attacking stand with Conway, who struck seven fours and five sixes in his 49-ball knock, but fell to Marsh.
Left-arm pace bowler Khaleel Ahmed denied Conway, who made a match-winning 85 not out against Sunrisers Hyderabad last Sunday, his hundred after getting the batter caught behind.
The 40-year-old Dhoni smashed an unbeaten eight-ball 21 to roll back the years.
Delhi were never in the chase after opener Srikar Bharat (8) and David Warner (19) returned to the dugout inside five overs.
Moeen soon sent back Mitchell Marsh for 25 and then struck twice in one over including the wicket of skipper Rishabh Pant for 21.
The Delhi batting soon caved in as wickets fell like nine pins and bravo wrapped up the innings with two successive wickets.