David Moyes’ side were booed off at half-time after James Justin put Leeds ahead at the Hill Dickinson Stadium.
But Barry eased Everton fans’ frustration with a composed finish 14 minutes from full-time.
The 23-year-old has scored four goals in his last five league games after netting just once in his first 18 following his move from Villarreal last year.
Everton are in 10th place, while Leeds sit six points above the Premier League relegation zone.
Wins for West Ham and Nottingham Forest at the weekend had increased the pressure on Leeds to counter with a victory of their own as the relegation battle heats up.
With that in mind, blowing the lead was a missed opportunity for Daniel Farke’s team, who have lost only one of their last 10 league matches.
Farke had described Dominic Calvert-Lewin as “one of the best English strikers around” ahead of the Leeds striker’s first return to his former club.
Calvert-Lewin has defied his critics since joining Leeds in August, scoring eight goals in his last 10 league matches after netting just three times in the final season of an injury-plagued spell with Everton.
But Calvert-Lewin was unable to haunt Everton in his return to Merseyside.
While Everton were coming off an impressive 1-0 win against title contenders Aston Villa, they struggled to find any momentum without injured England forward Jack Grealish, who is out for several months with a broken foot.
Leeds went ahead with a well-crafted move in the 28th minute.
Anton Stach crossed towards Calvert-Lewin and James Tarkowski was unable to clear, allowing Justin time to pounce with a clinical finish from 10 yards.
Calvert-Lewin was inches away from doubling Leeds’ advantage moments later when he prodded Jayden Bogle’s cross against the post from close-range.
Fresh from winning the Africa Cup of Nations with Senegal, Iliman Ndiaye had a chance to haul Everton level after the interval but Sebastiaan Bornauw made a superb last-ditch block.
Barry went close with an audacious flick that forced a brilliant save from Leeds keeper Karl Darlow.
It was a warning Leeds failed to heed as Barry hauled Everton level in the 76th minute.
Barry made a perfectly timed run to meet Idrissa Gueye’s low cross, clipping a deft finish over Darlow from six yards.
Everton pushed for a winner and Gueye, another of Senegal’s Africa Cup of Nations heroes, almost got it when he thundered a fierce strike against the bar. (AFP)
