The letter, co-ordinated by Labour MPs Jess Phillips and Stella Creasy and signed by cross-party members and peers, reads: "We do not presume guilt, of course, but we believe due process should be followed and the [Swedish] complainant should see justice be done".
Assange was arrested on Thursday in London when Ecuador revoked his diplomatic asylum after seven years of being holed up its embassy.
Assange first sought asylum in the Ecuadorian embassy in 2012, when he skipped bail after an extradition order to Sweden, where someone had accused him of sexual assault.
The MPs note that the investigation in Sweden was "discontinued only because of Assange's unavailability" while he was inside the Ecuadorian embassy in London.
With Assange facing extradition proceedings and up to five years in federal prison on the United States computer hacking charge, shadow home secretary Diane Abbott told BBC Radio 4's Today programme that the UK should resist handing him over. British police said Assange had been arrested Thursday for breaching his bail conditions and in relation to the us arrest request.
Elisabeth Massi Fritz, who represents the unnamed woman, said news of Mr Assange's arrest was "a shock to my client" and something "we have been waiting and hoping for since 2012".
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And if the European Council did not agree on an extension at all, "there would be a risk of an accidental no-deal Brexit", he said.
Assange faces what is likely to be a titanic struggle to fend off extradition to the USA - and possibly a second extradition request from Sweden on rape allegations.
"He said: "I told you so", Robinson told reporters and supporters, including fashion designer Vivienne Westwood, outside court on Thursday.
If Britain receives competing extradition requests, lawyers say the Home Secretary would have some leeway in deciding which takes priority.
But WikiLeaks editor Kristinn Hrafnsson said he fears more charges will be added, meaning Assange could face decades in a United States prison.
Assange, 47, has denied the sexual misconduct allegations, which he claims are politically motivated. She said a request by Swedish authorities, however, shouldn't be ignored.
After Assange's arrest, Swedish prosecutor Eva-Marie Persson was tapped to look into a request from a lawyer for one of the accusers, to find out whether the case can be pursued.
Following his arrest, Assange, with grey hair tied into a pony tail and sporting a long beard, appeared at Westminster Magistrates' Court, where a judge accused him of behaving like a "narcissist".
Former Home Secretary Lord Blunkett said people giving Assange "unqualified support" were being "extremely unwise".