Aberdeen produced their worst footballing display of the season in the first half, with several defensive lapses and blunders giving Hearts total control of the first 20 minutes.
And it was during this opening spell that the home side took the lead. With just three minutes on the clock, Alim Ozturk climbed at the back post to win a Sam Nicholson corner, and Paterson guided the ball into the net with his head.
The Dons' defence was again in disarray six minutes later as they failed to clear a corner and the ball slithered across the penalty area beyond the clutches of Danny Ward, but Blazej Augustyn could only stab his effort into the side-netting.
Aberdeen had an opportunity to respond when Miguel Pallardo chopped down Graeme Shinnie, but Niall McGinn's subsequent free-kick from 25 yards was straight into the arms Neil Alexander.
Hearts again drove forward and Arnaud Djoum tried his luck from 25 yards, but this time Ward was more assured in dealing with the situation.
Augustyn rose to meet another set-piece at the back post, and although Osman Sow could not create enough space for a shot, he laid a pass back to Ozturk on the 18-yard line, but Kenny McLean got across to divert the ball wide for a corner.
McLean again came to Aberdeen's rescue when he rose inside the six-yard box to block a net-bound header from Augustyn.
Tempers were becoming frayed amid the noise at Tynecastle and referee John Beaton booked Aberdeen striker Adam Rooney and Hearts defender Igor Rossi when they clashed
Nine minutes after the break Shay Logan took the legs from Nicholson just outside the penalty area and Ozturk crashed the resulting free-kick straight into the arms of Ward.
Two minutes later, Rossi appeared to pull Rooney down with his arm inside the Hearts box as the Irishman tried to connect with a lofted pass from Andrew Considine, but Beaton decided to wave play on.
Rooney then pounced on hesitant defending from Paterson and tested Alexander with a low drive to the 'keeper's near post, when a pass to Peter Pawlett may have been a better option.
Hearts survived another scare on 76 minutes when they struggled to clear a free-kick and Jonny Hayes drove into the box and sent a low drive towards the back post which went through the legs of Rooney to Considine, but the big defender scooped the ball over the bar from six yards.
The game had opened up and Hearts almost struck on a counter-attack with Gavin Reilly and Prince Buaben combining to set up Sow, but his low drive was comfortably saved by Ward.
And Ash Taylor got across to divert a half-volley from Sow wide after Mark Reynolds had misjudged a lofted pass from Paterson and allowed the ball to bounce into the box.
Ward raced out to nick the ball off the toe of Billy King as Sow released the substitute on a counter-attack from an Aberdeen corner.
Aberdeen came close in the final seconds of regulation-time when a snap shot by McGinn from inside the box was blocked by Alexander at his near post.
A minute later Reynolds won a deep corner and Taylor diverted the ball towards goal, but Alexander got his hands up to save under his bar as Hearts held on to secure a place in the last 16.
Source: Sky Sports