In Saturday's semi-finals at the Romai Tennis Academy, No 3-seeded Halep defeated countrywoman Alexandra Cadantu 6-2, 7-6 (7/1), while Meusburger upset No. 6 Chanelle Scheepers of South Africa 6-2, 6-2.
Halep won the first set in 27 minutes and managed several shots very close to the lines in the tiebreak to win the match.
"I played very well in the first set," Halep said.
"It was a good idea to move her between the sides as she returns balls very strong from the middle."
Meusburger has yet to win a WTA tournament and this will be her second final after Bad Gastein in 2007.
Results from the WTA Budapest Grand Prix (x denotes seeding):
Semi-finals:
Simona Halep (ROM x3) bt Alexandra Cadantu (ROM) 6-2, 7-6 (7/1)
Yvonne Meusburger (AUT) bt Chanelle Scheepers (RSA x6) 6-2, 6-2