The reserve-team player responded to Evans's five-year prison sentence by tweeting various comments in relation to the case and his support for his team-mate.
A complaint about the 19-year-old's tweets was made to North Wales Police by a leading rape charity.
In a statement, the club said: "Sheffield United Football Club have suspended reserve-team player Connor Brown with immediate effect pending an internal investigation into comments apparently posted on his personal Twitter account.
"There will be no further comment from the club during the process of the investigation which it hopes to conclude swiftly."
Meanwhile, North Wales Police are conducting an investigation after the victim of the rape was named and subjected to abuse on Twitter and other websites.
Victims of rape and all other sexual offences are automatically entitled to lifelong anonymity.
It is a criminal offence punishable on summary conviction by a £5,000 fine to identify the victim of a sexual offence in a publication or programme.
Police said yesterday they were "collating" website posts and would deal "robustly" with social media users who allegedly named and abused the woman.
Wales international Evans, 23, was jailed at Caernarfon Crown Court on Friday after being found guilty of raping the woman at a hotel in North Wales last May.
He admitted having had sex with her, but the woman told the jury that she had no memory of the incident - and the prosecution argued that she was too drunk to consent to sexual intercourse.
The striker, who hit 35 goals in 42 appearances in all competitions for Sheffield United, has indicated that he wants to appeal against the conviction.
Port Vale defender Clayton McDonald, 23, who also admitted having sex with the victim, was found not guilty of the same charge.