Bojan was a late replacement for Sweden striker Zlatan Ibrahimovic who injured his calf in the warm-up but was not missed as a weakened Barca swept aside Bilbao.
Jeffren, 22, opened the scoring after 27 minutes and Bojan, 19, a fellow product of the club's youth system, struck either side of half time to put the European champions in control.
"I suppose you were shaken when you read the team sheet," Barca coach Pep Guardiola joked to reporters. "But it is possible to go to the end of the world with these players."
Guardiola had Tuesday's Champions League quarter-final return leg against Arsenal and a top-of-the-table La Liga clash at Real Madrid next Saturday on his mind when he put out an unfamiliar side at the Nou Camp.
Jeffren stroked the ball in at the back post from Eric Abidal's low cross and it was Carles Puyol's interception and pass that set up Bojan for the second before the break.
Bojan struck a fierce third from the edge of the area after a sweeping move in the 59th minute before Messi slid in Pedro's cross to secure victory.
The victory came at a cost to Barca though. Injured Ibrahimovic was ruled out of the Arsenal game and is a doubt for the trip to the Bernabeu, as is defender Gerard Pique who has pulled a thigh muscle.
