The Bosnia-Herzegovina midfielder - signed from Hungarian side Ferencvaros last Monday - was brought on after half-time on Sunday as Everton stepped up their preparations for the new Premier League season.
But the 21-year-old's debut was not marked with a win, as an entertaining affair - which was played as Everton midfielder Leon Osman's testimonial to mark his 10 years at the club - ended in a draw.
A mistake from Porto goalkeeper Fabiano gifted Steven Naismith the opener in the 42nd minute, but that strike was cancelled out by Porto's Colombia international Jackson Martinez 12 minutes after the break.
Everton shot-stopper Tim Howard was given a run-out after returning from his World Cup exertions with the United States, but his cameo lasted only 19 minutes, with Joel Robles brought on in his place.
After a first-half scarce on chances but largely dictated by Porto, Everton took the lead in fortuitous fashion.
Fabiano's dismal clearance went straight to Aiden McGeady, who played in Naismith to present the Scot with the simple task of finding the bottom corner.
Besic was brought on at the interval for Ross Barkley, while Osman did his best to create opportunities as the hosts pressed for a second.
But Porto's quality up front told when forward Martinez levelled with a neat to finish to round off a well-worked move from the Portuguese side.
Both teams spurned chances in the quest for a winner, Porto playmaker Juan Quintero seeing an effort deflected narrowly wide of the post before Osman and defender Tyias Browning went close for Everton.
But neither side could snatch a second goal, meaning an evenly-matched contest finished all square.
