He appeared before 15,000 activists in Jackson, Mississippi, being introduced by and sharing the stage with Mr Trump.
And he said the party could "beat the pollsters" in the presidential race.
Mr Trump, who is trailing his rival Hillary Clinton in the opinion polls, backed the UK's exit from the EU.
In a tweet last week, Mr Trump said: "They will soon be calling me Mr Brexit."
Mr Trump introduced Mr Farage as the man who "brilliantly" led the UK Independence Party's campaign to secure a vote on the future of the UK's 40-year membership of the European Union.
Mr Farage began his address by saying he had a "message of hope and optimism" for the Republican Party.
He drew on parallels between Mr Trump's bid for the White House and that of the Brexit campaign's "people's army of ordinary citizens", which he said engaged successfully with the public prior to the UK's referendum vote on whether to leave the EU.
He told those gathered: "If you want change in this country, you better get your walking boots on, you better get out there campaigning.
"And remember, anything is possible if enough decent people are prepared to stand up against the establishment."
Mr Farage also said the Republican campaign represented a "fantastic opportunity".
He added: "You can beat the pollsters, you can beat the commentators, you can beat Washington."
Mr Farage, who attended the Republican convention in Cleveland last month, had previously said he would not "fall into the trap" of personally endorsing Mr Trump in his quest to reach the White House.
However, during his rally appearance he said that if he were an American, he would not vote for Mrs Clinton "even if you paid me".
By Gavin Hewitt, in Jackson, Mississippi
Donald Trump showered Nigel Farage with praise as the leader who had helped Britain regain control of its borders.
The two men shook hands and then Mr Farage spoke for 10 minutes. He was greeted with strong applause.
He said there were parallels between Brexit and what was happening in the US - little people let down by global corporations. "We made June 23 our Independence Day when we smashed the establishment", he told a crowd waving "make America great again" placards.
From the side of the stage Mr Trump smiled and clapped. Mr Farage spoke about when President Obama came to the UK and urged voters to remain in the EU. "He talked down to us," he said, "he treated us as nothing."
Mr Farage said "I can't possibly tell you how you should vote", but added he would not vote for Hillary Clinton "if you paid me".
It was not an endorsement but it came close.
The UKIP leader earlier told local radio in the state that the similarities between Brexit and the US election were "uncanny".
Speaking on Super Talk Radio in the US state, Mr Farage said he had been part of a "political revolution" in the UK and there were the makings of a similar movement in the US.
He compared the federal government in Washington DC to the European Commission, saying many people felt it had become "its own country", and claimed the Democrat nominee, Hillary Clinton, epitomised the status quo.
"I'm going to say to people in this country that the circumstances, the similarities, the parallels, between the people that voted Brexit and the people that could beat Clinton in a few weeks' time here in America are uncanny."
He suggested it did not matter that the political establishment, including many top Republicans, were shunning Mr Trump's campaign.
Mr Farage, who is credited with securing the referendum on the UK's membership of the EU and helping to pull off the surprise Leave victory, is standing down next month as the party's figurehead, but will remain as a member of the European Parliament.
However, Mr Farage's association with Mr Trump has not gone down well with some senior members of UKIP, including its sole MP Douglas Carswell, who tweeted "it's all going a bit South Thanet for the US Republicans" - a reference to Mr Farage's failed attempt to win a seat in the Kent constituency last year.
And Suzanne Evans, the former UKIP policy chief currently suspended by the party, suggested Mr Farage was trending on Twitter "for all the wrong reasons".
Lord Sugar, a former member and significant donor of the Labour Party who backed the Remain campaign, tweeted Mr Farage was a "creep" by claiming he won the Brexit vote.
Outgoing UKIP leader Nigel Farage has urged Republicans to "get your walking boots on" and drum up support for Republican presidential nominee Donald Trump.