Callum Paterson and Igor Rossi both headed in early goals to give Robbie Neilson's side a dream start.
Alim Ozturk stabbed in a third before the break but sliced into his own net to give Infonet one back before Rossi restored Hearts' three-goal lead.
Vladimir Voskoboinikov fired in a second for the home side.
That gave Hearts a 6-3 aggregate win and means they will face Birkirkara of Malta next week.
Scotland full-back Paterson was deployed as a wide forward and he showed his attacking prowess by giving the visitors the lead within the opening minutes when he bulleted Jamie Walker's corner into the net from six yards.
With the home side struggling to cope with Hearts' set-pieces, it was soon 2-0 when Walker this time found Rossi at the back post with a free-kick.
Conor Sammon passed up a golden opportunity to claim his first goal in Hearts colours after meeting an Arnaud Djoum delivery; the Irishman's header drifting wide.
On the stroke of half-time, another magnificent Walker corner-kick found the head of Rossi, who nodded the ball down for Ozturk net at the back post.
Infonet claimed a well-worked consolation after the break when Aleksandr Volodin surged down the right flank and delivered a super low cross, which Ozturk could only turn into his own net.
Those celebrations were short-lived, however, as the hosts were again found woefully incapable of defending a Walker free-kick 60 seconds later, allowing Rossi to blast home in the ensuing scramble.
Infonet continued to press and reduced arrears on the hour mark when Voskoboinikov volleyed home at the back post before Jevgeni Harin hit the crossbar, but Hearts were already well on their way to victory.
Hearts cruised into the second qualifying round of the Europa League with a comfortable win over Infonet Tallinn in Estonia.