Barna traded break points with Bychkova in a tough first set before cruising through the decider to book her place in the final eight.
"It is always good to win," Barna said, acknowledging the first set had been a tough fight.
Marrero lost the first set Poutchek before dominating the rest of the match.
In other play Wednesday, sixth-seeded Anna-Lena Groenfeld defeated fellow German Antonia Matic, 6-4, 6-4, while Italy's Antonella Serra Zanetti advanced to the second round after Julia Schruff of Germany retired injured with the scores at 6-3, 3-0.
Barna said she hoped to play Zanetti in the quarterfinals and not Iroda Tulyaganova of Uzbekistan, the hometown favorite who upset top-seeded Marion Bartoli of France in first-round play Tuesday.
CA Trophy
Top-seeded David Nalbandian of Argentina had to battle harder than expected, but still managed to beat qualifier Andreas Seppi of Italy 6-4, 6-3 during the US$845,908 tournament.
The former Wimbledon finalist profited from two unforced errors in the eighth game of the first set to earn the decisive break.
Nalbandian will next face Stefan Koubek of Austria, who defeated Alberto Martin of Spain 7-5, 6-1.
Also, Olympic champion Nicolas Massu of Chile advanced to the second round for the first time in four tries, edging Karol Kucera of Slovakia 6-4, 6-7 (5), 6-4.
Following early exits the last three years, Massu said he was "all set for success this time."
But despite having seven break points, Massu managed only one break in the first set.
In the second set, Massu broke Kucera early to take a 3-1 lead, but the Slovak broke back in the seventh game as Massu started to struggled.
Massu, who won the Olympic singles and doubles titles in Athens, then closed out the match after breaking Kucera in the third set.
"It was very difficult, but you cannot choose how your opponent is playing," Massu said. "I did not play well at the end of the second set, so Karol got a chance to come back."
Massu will next play South African qualifier Rik de Voest.
Other winners included Tommy Haas of Germany, Luis Horna of Peru and David Sanguinetti of Italy.
Haas, the 2001 champion, ousted Wesley Moodie of South Africa 6-3, 6-0, while Horna topped David Ferrer of Spain 6-3, 6-2 and Sanguinetti edged Mariano Zabaleta of Argentina 7-6 (7), 7-5.
Open de Moselle
Top-seeded Ivan Ljubicic of Croatia downed Victor Hanescu of Romania 6-4, 6-4 in the first round of the US$446,050 tennis tournament on Tuesday.
Defending champion Arnaud Clement of France, countryman Jerome Haehnel and Alex Corretja of Spain also won.
Clement, a former Australian Open finalist, downed eighth-seeded Gilles Muller of Luxembourg 6-4, 7-6 (3), while Haehnel advanced to the second round by beating Jose Acasuso of Argentina 6-3, 7-6 (2).
Corretja, the French Open runner-up in 2001, defeated fourth-seeded Oliver Rochus of Belgium 6-3, 2-6, 7-5.
Also, Florent Serra of France beat Juan Monaco of Argentina 6-4, 4-6, 6-0 and Marc Gicquel of France defeated countryman Gilles Simon 6-2, 2-6, 7-6 (5).
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