The 12-time Grand Slam champion saw off the American 6-3 6-1 6-2 to set up a meeting with Argentina's Horacio Zeballos in the next phase.
Djokovic, who finished runner-up at this year's Queen's, had things his own way for much of the contest, keeping the world number 57 at bay with his powerful drop-shots.
The Serb stuttered briefly in the fourth game of the third set when Sandgren broke his serve, but Djokovic was already two breaks up by this point and he went on to serve out the match.
The 31-year-old is seeking to bounce back from the disappointment of Wimbledon 2017, in which he was forced to retire from the quarter-final due to an elbow injury.
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