The hosts went into the break one goal in front thanks to captain Suso Santana's 26th-minute strike.
Their lead was doubled not long after half time through Diego Ifran.
Things got worse for Barca B late on as Patric was shown a straight red card with 10 minutes left.
It is a third home win of the season for Tenerife, who climb four places up to 17th in the table. Barca B stay ninth and miss out on the chance to climb into the top six.