The last time the sides met in December the Dons were hammered 5-0 at Tynecastle, a result which left them at the foot of the table, but this was a much tighter affair.
The hosts created the better chances but found Hearts goalkeeper Marian Kello in fine form, the Czech making two excellent saves to deny Sone Aluko and Rory McArdle in the first half.
The Tynecastle outfit enjoyed more of the play in the second half but rarely troubled Jamie Langfield in the Aberdeen goal.
The Jambos made five enforced changes from the side which defeated Dundee United last weekend as a result of Ian Black and Ruben Palazuelos missing out through suspension and Lee Wallace, David Obua and Suso Santana being out injured.
Andrew Driver, David Templeton, Ryan Stevenson, Craig Thomson and Adrian Mrowiec were all drafted in as replacements.
Paul Hartley returned to the starting line-up for the Dons after being benched for the 1-0 win over Hamilton on Tuesday.
The former Hearts midfielder replaced Derek Young, while Andrew Considine came in for the absent Zander Diamond.
The visitors had their goalkeeper Kello to thank for preventing them going a goal behind inside two minutes when the Czech stopper showed magnificent agility to tip a curling Aluko effort over the bar.
Hearts had a chance to test the Aberdeen rearguard, playing for the first time this season without Diamond, after 12 minutes but Rudi Skacel could not divert his header from Thomson's corner on target.
Dons defender McArdle had to leave the field for several minutes for treatment on a head injury and he almost broke the deadlock on his return with a downward header that was tipped round the post by Kello.
The Jambos were forced to make a change after 30 minutes when Andy Webster limped off and was replaced by Ismael Bouzid. The French-born Algerian had only been on the park for three minutes when he received the first booking of the match for a late tackle on Steven Smith.
Stephen Elliott flashed an effort just wide of Langfield's left-hand post five minutes before the break in a rare Hearts attack.
Hearts signalled their intent soon after the restart with Skacel's blistering strike kept out by a terrific save from Langfield before Marius Zaliukas headed over from a Thomson corner.
Aluko should have done better than shoot straight at Kello when he was fed in by Ryan Jack as the game continued at a relentless pace.
The visitors introduced Calum Elliot after 62 minutes for Driver, who was making his first start of the season, while the Dons replaced McNamee with Young.
The impressive Jack weaved through the Hearts defence with less than 15 minutes remaining but his eventual effort could only find the sidenetting.
The Dons brought on Nick Blackman and Josh Magennis for Scott Vernon and Aluko as they searched for a late winner.
Hearts had a late appeal for a penalty turned down when Templeton fell under pressure from McArdle as the match ended in a stalemate - with both sides missing out on the chance to secure a third successive victory.