Kink rifled in a piledriver from 25 metres and Vassiljev beat Serbia goalkeeper Vladimir Stojkovic with a stinging low shot from the edge of the area after central midfielder Zdravko Kuzmanovic had fired the hosts ahead on the hour.
Serbia's Aleksandar Lukovic scored a comical own goal in the 90th minute to compound a miserable debut for their new coach Vladimir Petrovic, who took over from Radomir Antic last month.
Serbia winger Milan Jovanovic said the hosts were left "stunned" by the defeat.
"To say that we are stunned is an understatement and this is surely the most shocking defeat of my career," Jovanovic, who was substituted at halftime, told reporters.
"We thought we were home and dry after we took the lead and nothing indicated the meltdown which followed, I am at a loss for words to explain what happened and why we suddenly crumbled."
The visitors always looked more dangerous than a despondent Serbian team, whose four-man defence struggled against a lively and creative Estonian midfield.
Centre-back Nemanja Vidic said he was confident Serbia could bounce back under the pressure of needing a result in their next match away to Italy on Tuesday.
"Estonia made it difficult for us with their tight defence and packed midfield, they surprised us and maybe we can do the same against Italy in Genoa because one defeat doesn't mean we are no longer a good team," he said.
PREDICATBLE STRATEGY
Estonia dominated for long spells and easily thwarted Serbia's predictable strategy of feeding lone striker Nikola Zigic with long balls from their own half.
Serbia captain Dejan Stankovic came close with an early header but the home side then faded away while the Estonians looked dangerous every time they came forward.
Stojkovic pulled off a close-range save to deny Estonia's Sergei Zenjov and Jovanovic shot high and wide from a good position at the other end before Kuzmanovic gave the hosts the lead with a deflected shot from 15 metres.
The home fans were silenced three minutes later when Kink's screamer left Stojkovic rooted to his line and Estonia struck again in the 73rd minute when Vassiljev was left unmarked to power his shot into the far corner.
Lukovic then compounded Serbia's woes when he headed a back-pass into his own net.
"We have to gird together now and produce our best performance against the Italians, it is a match we can't afford to lose," said Serbia winger Milos Krasic, who is likely to face some of his Juventus team mates on Tuesday.
"Things that were going on after the World Cup have obviously affected us," he added, referring to an acrimonious split between Antic and the Serbian Football Association.
Italy top the group on seven points with Estonia on six. Northern Ireland, Serbia and Slovenia have four with the Faroe Islands pointless.
