Driver lashed home the 24th-minute opener on his first appearance for more than six weeks, after a knee injury, and Gary Glen soon turned home the second.
Hearts should have put the Scottish Premier League clash far beyond Hibernian but Derek Riordan's 79th-minute goal set up an interesting finale.
The hosts held on without serious danger though as manager Jim Jefferies enjoyed a winning return to the fixture in his second spell as Hearts boss.
Jefferies took a risk by handing Driver only his second start since the teams drew at Easter Road on November 7, but it paid off in style as Hearts secured an important victory amid pressure from St Johnstone for a top-six place, while cutting the gap on their city rivals to seven points.
Hibernian were without skipper Chris Hogg as they missed the chance to move above Motherwell in the race for Europe with their recent record now reading just one win in seven SPL games.
Sol Bamba made a solid start in place of Hogg in central defence and Hearts' injury problems were exacerbated when centre-back Dawid Kucharski went off in the 18th minute.
Suso Santana came on at right midfield and, with Jose Goncalves absent through injury, Lee Wallace moved alongside Eggert Jonsson in central defence with Ruben Palazuelos filling in at left-back.
But Hearts soon took the lead when Suso's cross hit off the outside of the post and fell for Driver to volley into the roof of the net from 10 yards.
Hearts immediately made another change with Ryan Stevenson replacing Black after the midfielder failed to shake off a head injury.
But they doubled their lead when Glen nodded home from close range after Eggert Jonsson had headed on Driver's corner. Hearts soon twice went close from the same source but John Rankin cleared Jonsson's volley off the line and David Obua headed over from eight yards.
Hibs looked shaken and they survived another scare from a Driver corner when Palazuelos failed to turn a bouncing ball on target from close range.
Hibs continued with Anthony Stokes as a lone striker and the second half did not start any better for them with Hearts missing two chances inside 90 seconds.
Obua hooked over Jason Thomson's cross and Suso missed an open goal from inside the six-yard box after Obua's low cross had taken out Graham Stack, although the Spaniard was under intense pressure from two opponents.
Glen missed another glorious chance in the 53rd minute when Obua's return pass played him clean through but the youngster over-ran the ball as he went round Stack.
Hibs brought on Abdessalam Benjelloun for Rankin to move to a 4-4-2 formation but Glen went close again when he found space to glance Kingston's free-kick towards the corner of the goal, with Stack stretching to push it away.
Hibs finally began to push Hearts back but the hosts remained resolute and Driver nearly netted a brilliant goal after flicking the ball over Ian Murray and hooking it past Stack, only for Lewis Stevenson to clear off the line.
Benjelloun shot over from 20 yards and Hibs did not look like getting back into the game, but Riordan netted out of the blue.
Several Hearts players missed the chance to clear David Wotherspoon's low cross from the right and Riordan fired the ball in off the bar at the back post.
But Hibs were unable to take advantage of their avenue back into the match and Stack saved from Suso and Stevenson as Hearts comfortably held on.