With both Celtic and Rangers in action on Sunday, Derek McInnes' men had the chance to apply pressure to the Glasgow giants in a tight three-way title race.
Hearts needed a reaction from last weekend's 4-2 defeat at Rangers to keep their quest to become the first non-Old Firm champions of Scotland in 41 years on track.
Chesnokov, a January signing from Kazakh side FC Tobol, prodded home the only goal just before half-time.
The home side were kept at bay by Falkirk goalkeeper Scott Bain in the second half, but did enough for a vital three points.
Second-placed Rangers can cut the gap back to two points when they travel to bottom-of-the-table Livingston.
Defending champions Celtic are now six points off the top but have two games in hand, starting with the visit of Hibernian on Sunday.