Reyes' ninth home run broke a 2-all in the fourth inning and sent the Blue Jays to their third win in a row.
Jose Bautista's streak of homering in five straight games ended, but he had two singles and an RBI for Toronto.
Dickey (11-12) gave up two runs, struck out six and walked three.
Reyes, who singled and scored in the third, had his 13th multiple-hit game in his last 29 games.
Hellickson gave up five runs on eight hits in 3 1-3 innings, the shortest of his nine starts since coming back from January elbow surgery.
A win on Wednesday or Thursday night would give the Blue Jays their first series victory at Tropicana Field since April 6-8, 2007.
Blue Jays: INF Brett Lawrie (oblique) was transferred from the 15- to 60-day disabled list, which ends his regular season. ... RHP Brandon Morrow (finger) and 1B Dan Johnson (hamstring) were activated from the disabled list. ... LHP Brett Cecil was unavailable due to some minor stiffness.
Rays: CF Desmond Jennings (sore left knee) was out of the lineup for the fifth straight game.
Rays RHP Chris Archer (8-7) and Blue Jays RHP Marcus Stroman (8-5) are Wednesday night's scheduled starters. Archer is coming off a loss to Boston Friday where he gave up a season-high eight runs in four innings.
Former Blue Jays pitcher and one-time Rays bat boy Jesse Litsch threw the ceremonial first pitch. The St. Petersburg native, who announced his retirement last month, went 27-27 over parts of five seasons with Toronto.
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