Nagal lost to Frenchman Pierre-Hugues Herbert, a top-100 player, 5-7, 0-6 after failing to convert two break chances in the first set when the match was evenly poised at 5-all and Nagal was leading 15-40.
The 136th-ranked Nagal, who is aiming to get into the top-100 on the ATP rankings, will now take on Italy's Thomas Fabbiano for a place in the main draw of the tournament, which is one of the most important events in Spanish tennis.
Travaglia, seeded second in qualifying, dropped just three points on first serve to crush the 23-year-old Indian in 58 minutes on Saturday.
The gutsy show earned Nagal 45 main draw ranking points that will push him to 132 from the current 150th slot.
Nagal will play Spain's Albert Ramos-Vinolas, in what will be the Indian's maiden quarterfinal appearance at the ATP Tour level.
Nagal, India's second-highest ranked men's singles player, lost 2-6, 5-7, 3-6 to Berankis in a match that lasted two hours and six minutes.
Ankita Raina, who was waiting to know if she gets 'lucky loser' spot in the women's singles, could not make the cut.
Nagal has been awarded a wild card entry into the hard court major and he will come to know about his opponent when the draw is announced on Thursday.
Nagal, who does not give up easily, suffered a rather tame 2-6 2-6 defeat in the USD 320775 ATP250 event.
Nagal, who is India's second highest ranked men's singles player and holds a world ranking of 136 and Chinese world no. 123 Wang Xiyu are the recipients of the Asia-Pacific wildcard.
Seeded 16th Sumit loses 6-7 (4), 5-7 to Germany's seasoned player Dustin Brown while Prajnesh gets the better of Turkey's Cem Ilkel 6-3, 6-1.
Dominic Thiem beat Sumit Nagal in straight sets to advance to the third round of US Open.
Up next for Sumit Nagal will be Austria's top player Thiem, who prevailed against Spanish rival Jaume Munar 7-6(6) 6-3.
It will be second straight US Open singles main draw for Nagal, who came through the qualifiers last year and played a memorable match against iconic Roger Federer.
Sumit Nagal, who lost to world number 17 and three-time Grand Slam winner Stanislas Wawrinka in the singles, also lost his doubles match to bow out of the tournament.
The 23-year-old Indian bagged the first set 6-2 in stunning fashion after breaking the Swiss heavyweight's serve to raise hopes of an upset win.
Sixth seed and world number 127 had to battle hard to prevail 5-7 7-6(4) 6-3 in two hours and 21-minute second-round contest of the Euro 137,560 clay-court event.
The 22-year-old Indian, playing his first competitive match since his Davis Cup encounter against Croatia's Marin Cilic in March, had received a bye in the opening round.
Nagal is the only Indian in the men's field with Prajnesh Gunneswaran, ranked 132, missing out.
Nagal, who is currently India's highest-ranked singles player at 127, recently defeated second seed German Daniel Masur 6-1, 6-3 in the final match at the tournament.
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