The value of the deal has not been not disclosed.
The acquisition will add more than 60 employees and more than 6,000 fleet assets to Aggreko's Rental Solutions business.
Glasgow-based Aggreko said it planned to fully integrate Dryco into its North America business.
It also intends to develop "expansion plans to enter additional sectors and global markets".
Dryco specialises in moisture control, drying, heating, and cooling applications for a number of industries, including shipping, manufacturing, food processing and construction.
The company, which operates 10 full-service locations in North America, posted revenues last year of $19m (Â£14.6m).
Aggreko chief executive Chris Weston said: "This acquisition combines the deep technical strength of two highly respected market leaders and provides our customers with an unparalleled range of solutions that can address their HVAC (heating, ventilation and air conditioning) and moisture control needs.
"It also gives Aggreko an opportunity to further support our rapidly expanding customer base within the shipping, food and beverage and painting and coatings industries."
Temporary power suppler Aggreko is expanding its offering in the United States by buying industrial climate control equipment specialist Dryco.