The lanky left-hander was drafted into the home side after Tharanga suffered a hairline fracture in his right toe when being hit by a ball from fast bowler Lasith Malinga during Thursday's practice.
"The selectors have called up Vandort to take Tharanga's place for the first Test," Sri Lanka Cricket (SLC) spokesman Samantha Algama told AFP.
"Tharanga's progress will be monitored closely to see when he will be fit to play."
Vandort, who averages 51.22 in six Tests so far, is set to open the batting with debutant left-hander Malinda Warnapura when the first Test starts at the Sinhalese Sports Club here on Monday.
The 27-year-old Vandort, who played his last Test in England a year ago, made 38 in the three-day tour match against the Bangladeshis which ended in a draw here on Friday.
Sri Lanka will have a new pair at the top of the order after veteran Sanath Jayasuriya was rested for the series and former captain Marvan Atapattu opted out due to personal reasons.
Bangladesh will play three Tests and three one-day internationals on the five-week tour.