Infonet were rewarded for their bright start in the 21st minute when Jevgeni Harin thumped home a long-range volley to silence the home supporters on the night.
Hearts were level in the 28th minute, however, when Prince Buaben converted from the penalty spot after Vladimir Avilov had handled inside the box.
The home side then scored a second six minutes before the interval when Andrei Kalimullin turned into the back of his own net from close range after Callum Paterson had an attempt cleared off the line.
It was Infonet that looked the team most likely to register in the second 45 minutes, but Hearts managed to resist a spell of pressure to secure a first-leg lead in front of more than 14,000 spectators.
The second leg will take place in Estonia next Thursday night.