Lachie Turner and Cowan-Dickie both grabbed first-half braces, with Gareth Steenson kicking nine points as the hosts led 29-13 at the break.
Rob Buchanan went over in response and Tom Swiel converted having landing two earlier penalties.
Cowan-Dickie competed a treble before Charlie Walker and Danny Care replied and Tomas Francis saw red for Exeter.
Francis was sent off with eight minutes remaining, kicking out at a ruck and catching Care with his shin.
Defeat is Quins' second in three games, with their only triumph to date this season coming against newly-promoted Bristol. The loss was also compounded an injury to Jack Clifford, who left the field in the second half.
Quins led only briefly, thanks to the boot of Swiel as he landed two penalties after Turner's opening try.
Turner grabbed a second and Cowan-Dickie his first before Buchanan went in for Quins' next score.
Cowan-Dickie went over to secure a bonus-point before the break and another score by the England hooker helped put the game beyond the visitors' reach, despite tries from Walker and Care and a late numerical disadvantage.
Exeter director of rugby Rob Baxter: "We have been under a bit of pressure after two losses but it was fantastic to win and get a bonus point as well.
"We did enough and scored some good tries but we are frustrated by our second-half performance.
"We should have controlled territory a bit better than we did but I am pleased with the character shown by us.
"We were far from being at our best but I am not that upset that Quins came back into it. They are a successful Premiership side and they have got some dangerous players, who can score some good tries."
Harlequins director of rugby John Kingston: "We gave away two soft tries in the middle of the first half when we were 6-5 up.
"One of them was a simple overthrow and the second came from poor tackling from the first phase.
"It was very difficult to come back from there and we didn't start the second half as well as we would have wanted. It was a question of too little, too late and the bottom line is that we were punished for too many errors at key points.
"We have to alienate those errors, but Exeter did take every opportunity that came their way."
Exeter: Turner; Woodburn, Devoto, Slade, Whitten; Steenson (capt), Chudley; Moon, Cowan-Dickie, Holmes, Welch, Parling, Armand, Salvi, Waldrom.
Replacements: Yeandle, Hepburn, Francis, Atkins, Dennis, Lewis, Hill, Short.
Red card: Francis (71).
Harlequins: Brown; Yarde, Marchant, Stanley, Walker; Swiel, Care (capt); Marler, Buchanan, Sinckler, Twomey, Horwill, Robshaw, Wallace, Clifford.
Replacements: Ward, Lambert, Collier, Merrick, J Chisholm, Dickson, Lang, Alofa.
Referee: Matthew Carley.
Luke Cowan-Dickie scored a hat-trick as Exeter Chiefs beat Harlequins to record their first win of the season.