Hibs struggled to cope with the attacking play of Jamie Hamill and Stevenson down the right and the pair combined to open the scoring six minutes before the break.
Webster headed Hamill's 68th-minute corner over the line despite Akpo Sodje's attempts to clear as Hearts dominated the second period.
Hearts restored their experienced players after a youthful side secured a goalless draw at Tottenham in the Europa League with Scott Robinson and Gordon Smith spending the afternoon on the bench despite their impressive displays at White Hart Lane.
Hibs handed Leigh Griffiths a debut off the bench following his loan move from Wolves and Richie Towell came into the starting line-up after returning for a second loan spell from Celtic, but Hibs followed defeats by Kilmarnock and St Mirren with another loss.
Kick-off was delayed for several minutes as balloons and a smoke bomb were thrown from the Hibernian support.
There was little goalmouth action in a frantic first quarter, with Hearts unable to carve an opening despite having the edge. Hibernian winger Ivan Sproule was booked for a poor challenge on Danny Grainger before felling the same player with a clumsy challenge, but Craig Thomson was right to wave away calls for another card.
The action increased as the half progressed and the first decent chance fell to Garry O'Connor, who looked to have room to head home Sproule's cross but Webster recovered to challenge the striker and force him to send the header wide.
Andrew Driver quickly drew the first save when he dribbled past Towell and was handed a chance to shoot when Sproule tried to backheel a clearance after initially stopping the winger's run, but Graham Stack got down to save at his near post.
Hearts took the lead after Hibs lost the ball deep inside their half. Matt Thornhill waited for a pass to come to him in the left-back position and was dispossessed by Hamill, who exchanged passes with Ian Black and sent in a low cross that Stevenson swept home from six yards.
Thornhill almost made amends when he burst through the home defence but he could not curl the ball enough as he shot wide from 18 yards. Stevenson was booked for a hard and late tackle on Ian Murray and Hearts pressed hard for a second after the break.
Stack pushed away Elliott's angled drive after Stevenson headed on a long ball and Hibs had an escape when Elliot side-footed Stevenson's cutback against the post before the Hibernian goalkeeper made a superb recovery to clear Driver's follow-up off his line. O'Connor responded with a powerful swerving free-kick from 30 yards, which was well held by Marian Kello.
Stevenson set up another chance when he lofted a pass through for Hamill but the defender shot against Stack with Black screaming for a square ball.
Griffiths was booked for a late tackle on Hamill moments after coming on and Hearts lost David Obua to a head knock after he was caught by Stack as he beat the goalkeeper to Stevenson's cross and headed wide.
The pressure soon paid off. Webster headed Hamill's corner and substitute Sodje was unable to prevent the ball going over the line before clearing with Eggert Jonsson making sure anyway by forcing home the rebound.
Hibs looked frustrated and Murray and O'Connor were booked for fouls on Hamill either side of a yellow card for Sodje for catching Kello late as they contested a high ball.
Thomson handed the fourth official missiles, including a coin, immediately after Kello had received treatment in front of the away support.
Kello was eventually replaced by Jamie MacDonald, who held long-range efforts from Griffiths and O'Connor as Hearts comfortably held on.