Makoto Hasebe sent keeper Michael Esser the wrong way with a well-taken penalty in the 74th minute and Ante Rebic slotted in from close range after fine work from Alex Meier nine minutes later to seal their victory.
Darmstadt, in last place on nine points, defended bravely for more than an hour but eventually ran out of steam.
Eintracht, with a sixth win in their last nine home matches, moved up to third place on 35 points, one above Borussia Dortmund, 1-0 winners on Saturday against second-placed RB Leipzig, who have 42 points.
Leaders Bayern Munich increased their gap at the top to four points despite a 1-1 draw against Schalke 04 on Saturday.
Earlier on Sunday, Paraguay international Raul Bobadilla sneaked into the box to fire in a stoppage-time winner as visitors Augsburg twice came from a goal down to beat strugglers Werder Bremen 3-2.
Augsburg are 10th on 24 points with Werder in 15th spot on 16 points, the same as Hamburg SV, who occupy the relegation playoff spot in 16th place.
(Reporting by Karolos Grohmann; Editing by Ian Chadband)
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