Injury-hit youngster Vaughan had only been on the pitch for 40 seconds when he fired home Marouane Fellaini's cross at the far post with six minutes remaining before South Africa international Pienaar lashed in a stunner in the final minute.
And it was hard luck on the Clarets who had matched the Blues for long periods until Stephen Jordan got himself sent off on the hour for tugging back Pienaar.
The Blues flew out of the traps and in the second minute Leon Osman wasted a great chance when he failed to control Yakubu's cut-back in the box.
Blues keeper Tim Howard was called upon in the 20th minute when he produced a fine stop to divert Steven Fletcher's shot over the bar.
A minute later he kept out Wade Elliott's low drive but Everton immediately broke and Leighton Baines' cross was crashed against the post by Yakubu.
And 12 minutes before the break Diniyar Bilyaletdinov skipped inside Jordan, but saw his shot brilliantly saved by Brian Jensen.
But it was the Clarets who started the second-half brightest with Chris Eagles' 70-yard dash setting up Kevin McDonald but the young Scot blazed his shot over the bar on 52.
Five minutes later lifelong Evertonian David Nugent was denied by the post after quick-thinking from Elliott.
But Burnley were up against it on the hour when Jordan, who was sent off on his last visit to Goodison Park in February 2006 for Manchester City, tugged back Pienaar and received his second yellow card.
That gave Everton renewed impetus and Jensen made another great stop on 65 to keep out Yakubu's looping header.
But the deadlock was finally broken on 84 when Vaughan clinically bagged his first Everton goal since December 2007.
And the win was sealed in the final minute when Yakubu rampaged forward from the halfway line, before teeing up Pienaar to smash his second goal of the season past Jensen.