"We view with relief the serious movement in the prisoner swap file," Haniya said.
"We stress that we will not abandon our responsibility towards our male and female prisoners."
Saed was arrested in 2007 while crossing the Gaza-Israel border.
She had cited medical reasons for wanting to enter Israel, which accused her of plotting a suicide bombing.
Israel freed 19 other Palestinian women on Friday before receiving the two-minute tape of Shalit, who looked healthy and sounded coherent.
Until Friday, the only signs of life from Shalit have been several letters and an audio recording. Only one of the letters - written three months after his capture - has been released. It came out just last month.
Hopes raised
The prisoner release in exchange for the video has stirred expectations that a deal could soon be reached to release Shalit in return for a wider Israeli prisoner amnesty, but officials on both sides said there was still some hard bargaining ahead.
For Shalit's release, Hamas has demanded that Israel free hundreds of Palestinians from its prisons.
Currently, more than 10,000 Palestinian prisoners remain behind Israeli bars.
The Palestinian Liberation Organisation, the Palestinian Authority and Hamas all insist there can be no peace deal with Israel before all these prisoners are freed.
Israeli newspaper editorials urged speedier work to release Shalit, 23, who was captured by Palestinian fighters in a June 2006 cross-border raid.
"Let's complete the swap," said the editorial headline in Haaretz.
The mass-circulation Yedioth Ahronoth daily ran a banner headline quoting Shalit's own words from the video: "May I fulfil my dream of going free, at last."