Konta registered her first Grand Slam victory earlier in the week at Flushing Meadows against Timea Babos but the Australian-born Briton could not double up as she lost a tight three-setter on Court 7.
The 21-year-old, who was born in Sydney to Hungarian parents, only switched allegiances to Britain in May having lived in England since 2005.
And she looked liked joining compatriot Laura Robson in the third round of the final Grand Slam of the year when she easily claimed the first set against Govortsova.
The second set went her opponent's way before Konta stormed out of the blocks in the final set, only to suffer a collapse and fall to defeat before a deciding tie-break was required.
In the men's doubles, British pair Jamie Delgado and Ken Slupski beat Johan Brunstrom of Sweden and James Cerretani of the US 4-6 6-4 6-3 in their opening match of the tournament.
Robson teamed up with Israeli Shahar Peer in the women's doubles but dropped to a 6-0 6-3 defeat to Sabine Lisicki and Shuai Peng in the first round.
There was disappointment too for Heather Watson, who along with partner Marina Erakovic retired from their match with Russian sixth seeds Ekaterina Makarova and Elena Vesnina at 3-2 down in the first set.
And in the mixed doubles draw, Jonathan Marray, partnered by Czech player Vladimira Uhlirova, lost 6-1 6-3 to Varvara Lepchenko and Donald Young of the US.