After leading 2-0 with Riga's 11th and 23rd minute goals, Pune FC conceded a goal two minutes after the break with United's Hasan Al Moustafa finding the target.
The win, Pune's first after five matches, revived their I-League hopes as they jumped to third place with 22 points from 13 matches, while United SC remained on 18 points.
Riga utilised a Dhanpal Ganeshan's long ball and beat the offside trap and finish brilliantly to give the visitors an early opening.
United SC ace forward Ranty Martins, who is rumoured to be joining East Bengal, continued his dry patch, missing as many as two goals as the home side struggled to find an equaliser.
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The Nigerian was a fine position in the 18th minute after being set up by Deepak Mondal but failed to give the finishing touches. Pune FC doubled the lead five minutes later.
It was a perfect start for the hosts after the changeover when Lalkamal Bhowmick's free-kick created utter confusion in the Pune penalty area and Al Moustafa smartly put the ball in the net.