Montenegro, who launched their debut Euro campaign with a 1-0 triumph over Group G rivals Wales on Friday, sank Bulgaria with a 35th-minute goal.
Zverotic notched his first international goal when he unleashed a thunderous shot from outside the penalty area.
It was Bulgaria's second successive defeat in the competition, following Friday's 4-0 rout by England, and coach Stanimir Stoilov announced his resignation immediately after the defeat by Montenegro.
"I wanted the team to get to the finals but it's obvious that after two defeats, one at home, it's going to be very difficult," Stoilov told reporters.
"I've already informed the team and Bulgarian Football Union president (Borislav Mihaylov) of my decision (to quit)."
The home side were on top early but Ivelin Popov and Dimitar Rangelov wasted good chances, leaving the fans to rue all-time leading scorer Dimitar Berbatov's decision to quit the team in May.
Stoilov made two substitutions at halftime, bringing on striker Valeri Bojinov for the ineffective Rangelov and Stanislav Genchev for defender Zhivko Milanov.
However, it was Montenegro who enjoyed plenty of possession after the interval and Radomir Djalovic almost doubled their lead when he headed just wide.
At the other end, keeper Mladen Bozovic did well to parry Martin Petrov's curling free kick and Stanislav Angelov's rasping volley from the edge of the penalty area.
Petrov also hit the bar from a free kick while Veselin Minev went close in injury time.
Bulgaria are ranked 43rd by FIFA while Montenegro are down at number 73.
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