The Udinese striker, on loan at Watford, won the Championship player of the season award this week and celebrated by heading his country ahead two minutes after half-time.
Vydra has impressed English football fans with his performances in the second tier, where he has scored 20 goals, and his impressive 81st-minute strike was further evidence of the young Czech's burgeoning talent.
Substitute Daniel Kolar then scored to rubberstamp a win that moves Michal Bilek's men on to eight points in their bid to reach next year's World Cup, while Armenia remain in fifth place with three points.
Armenia's Edgar Manucharyan tested Petr Cech with a swerving shot before Czech defender Tomas Sivok scooped over the crossbar in the early exchanges.
Home keeper Roman Berezovsky fumbled a Jaroslav Plasil corner to safety before the break, but two minutes after the restart, the Czech Republic took the lead.
When Plasil lofted the ball across the box to Vydra at the end of a flowing move, the 20-year-old headed across a stranded Berezovsky.
And with nine minutes left to play it was Vydra who put the result beyond doubt with a splendid solo effort as he span his marker to smash the ball home from the edge of the area.
Armenia were beaten but there was still time for Kolar to convert Tomas Hubschman's left-wing cross five minutes into stoppage time.