Al Dawsari's perfectly placed spot-kick gave Al-Hilal the lead in the first minutes of the match after Matheuzinho tripped Vietto inside the box.
Pedro equalised for Flamengo with a close-range strike in the 20th minute but Al-Hilal regained the lead just before half time as Al Dawsari again converted from the spot, with Gerson sent off for a second booking after stomping on Vietto in the penalty area.
With Flamengo reduced to 10 men for the entire second half, Al-Hilal controlled the tempo and eventually scored the third, when Al Dawsari returned the favour to Vietto, putting the ball on a plate for the Argentinian to score with a point-blank strike.
Pedro's close-range range effort in added time was too little to late for Flamengo, who became only the sixth Copa Libertadores champions not to advance to the final.
Al-Hilal, the first Saudi side to reach the Club World Cup final, will face the winners of Real Madrid's semi-final against Al Ahly on Wednesday in Rabat.