The pair were on target in the second half at Goodison Park on Wednesday night as the Merseyside club took on the South American side named after them for the first time in an historic friendly.
Chile's Everton were founded in 1909 after the original Everton made a positive impression on a pioneering tour across the Atlantic.
The match was played for the newly-created Brotherhood Cup and could be the start of closer ties between the two clubs.
The main purpose of the exercise for Moyes, however, was fine-tuning ahead of the Premier League season.
The Scot will have been pleased with a lively display from new signing Beckford and a first run-out for Marouane Fellaini since injuring his ankle in a Merseyside derby in February.
Referee Mark Halsey, who is in remission from cancer, was also given a warm reception by an appreciative Goodison Park crowd.
The host Everton were a class above their opponents and squandered a number of chances to take the lead in a goalless first half.
Steven Pienaar, whose future remains the subject of speculation, had the first of them with a free-kick that was saved by Gustavo Dalsasso.
Beckford burst into the area and fired at Dalsasso before Tim Cahill headed at the goalkeeper and Pienaar flicked wide.
Beckford also missed the target when well placed and Cahill sent a glancing header across the face of goal.
Jack Rodwell saw a good volley blocked, Beckford fired narrowly wide and Cahill put another good header against the bar. Pienaar then tested Dalsasso with a long-range shot.
Two speculative efforts from Mauro Guevgeozian that flew over were the best of the visitors' first-half efforts.
English Everton claimed the lead after 50 minutes when Beckford met a cross from fellow new recruit Magaye Gueye with a looping header.
Chile Everton almost responded immediately as Guevgeozian shot straight at Jan Mucha but the Merseysiders soon doubled their lead.
Substitute Fellaini got involved by nodding down a Gueye corner and Bilyaletdinov blasted a low shot home.
Gueye went close to a third with a shot saved by Dalsasso and the goalkeeper then managed to scramble another Beckford attempt off the line.