Prior to Barkero's strike, Roberto Soldado had put Los Che in pole position for the win with a goal in the 48th minute.
Barkero scored thanks to a strong individual effort and put a left-footed shot into the lower right of the net from centre of the box.
Jonas was on the pitch for 78 minutes. He scored in the 27th minute, took three shots and completed 28 passes.
Jonas's favourite passing target was Daniel Parejo, with whom he linked up five times.
Valencia maintained control of the ball for 66.8% of the match, a testament to their movement and linking play in the open field.
Xotos got off 13 shots, including six on target, during the match, a direct result of their 60 duels and 14 headers won.
Soldado had three shots for Valencia, more than any of his team-mates.
The combination of Jeremy Mathieu to Víctor Ruiz was the most common during the match, as the two players connected 16 times.
Los Granotes enjoyed possession of the ball for a mere 33.2% of the contest.
Levante had eight shots, including four on target.
Míchel made the biggest impact of any Levante substitute after coming on for Pape Diop in the 58th minute, as he took two shots.