The DUP urged the Sinn Fein leadership to sanction the Foyle MLA for her "deeply offensive tweet". The SDLP called on Mary Lou McDonald and Michelle O'Neill to clarify their position on her tweet, which was later deleted.
In a damning High Court ruling last week Sinn Fein was ordered to stop blocking the implementation of the pension scheme. The party claims the criteria discriminate against former republican prisoners.
The Justice Minister branded Ms Anderson's tweet "an outrageous, gratuitous insult to all those who will qualify for the pension".
Mrs Long said: "Sinn Fein have a right to their view about how the Government has implemented the pension. They have a right to make their case, as has every party. They do not have a right to launch such vitriol at those who will qualify or smear them in this way."
SDLP leader Colum Eastwood described Ms Anderson's comments as "unacceptable, disgusting and grossly insulting to hundreds of victims who sustained life-changing physical and psychological injuries".
Calling for an apology, he said: "The campaign Sinn Fein has waged against delivering this pension, forcing victims who have endured decades of hardship living with deep physical and emotional scars to go to court to get what they deserve, has been deeply disturbing.
"How much more insult can Sinn Fein add to the injury that these victims have sustained?
"How much more pain are they willing to inflict on people who have survived the most horrendous atrocities? Martina Anderson is a senior Sinn Fein MLA. Her party president Mary Lou McDonald and joint First Minister Michelle O'Neill should urgently clarify if they believe victims applying for this pension are 'mainly those who fought Britain's dirty war' or were 'involved in collusion'."
UUP MLA Doug Beattie said: "Tweets like this confirm the complete lack of integrity from Sinn Fein.
"Having spent years denying the rights of innocent victims to a pension, and followed that up by disobeying the law, we now have this piece of deluded nonsense from Martina Anderson."
Mr Beattie said the party seemed determined to "slur" IRA victims and was showing it was shameless and had "an appalling lack of humanity".
Foyle DUP MLA Gary Middleton described Ms Anderson's comments as "patent nonsense".
He said: "The Sinn Fein leadership must deal with this shocking tweet.
"This cannot be allowed to go unchecked.
"People who have lived most of their lives with shrapnel from an explosion in their body - or who are haunted with the smell, taste and noise of a bombing - should not be labelled by Martina Anderson."
He added that he would be "exploring with the Northern Ireland Assembly authorities whether this statement is in fact a breach of the Members' code of conduct".
He said: "In my view it falls well short of decency, never mind the standards expected from an MLA."
TUV leader Jim Allister said that Ms Anderson could "always be relied on to lower the tone" but her "outbursts" likely reflected Sinn Fein's actual thinking better than the contributions of "some of its more subtle figures".
Asked to comment on Ms Anderson's tweet, Sinn Fein said: "The proposed victims' pension scheme is not the type of scheme envisaged under the Stormont House Agreement and is completely inconsistent with the legal and agreed definition of a victim.
"It's our view that this scheme will be exclusionary, discriminatory and divisive. Its policy intent was and remains to create a hierarchy of victims and reinforce the British state narrative around the conflict.
"Sinn Fein remain committed to delivering a pension scheme which is based on equality and open to everyone who was seriously physically and psychologically injured during the conflict."