Matt Kvesic grabbed a forward based score as Worcester looked to fire a famous home comeback before Josh Drauniniu's converted score put Warriors ahead on the hour mark.
But a yellow card to Josh Matavesi with ten minutes to go saw Chiefs strike again through Dean Mumm with three minutes left as Warriors had to settle for a losing bonus point on home soil.
Exeter looked to play from the first minute but an early miss pass wide right went to the touch line and not hands as Worcester looked to force an early platform.
Josh Matavesi and Neil Best looked to ride over tackles before David Lemi smashed past his opposite man and the recalled Semisi Taulava and Chris Jones combined to force a penalty and the opening points from Goode.
The Chiefs were adopting an attacking brand of running rugby from the start with Richard Baxter and Jack Yeandle making yards before sustained pressure saw Luke Arscott crash over from close range with Gareth Steenson adding the extras.
Visiting scrum half Haydn Thomas made a rare error as he kicked out on the full to give Worcester an attacking lineout. The home side looked to go wide at pace and Matavesi almost broke away before Lemi was pulled back in full flight for a forward pass.
A loose pass in midfield saw Pennell, Drauniniu and Grove combine with some sublime handling and when the ball was recycled it was Goode who picked out Lemi with a cross kick.
He touched down to huge celebrations, but the try was then chalked off by the touch judge for a knock on to disbelief inside Sixways.
Goode ate into the lead with a penalty, as Exeter were penalised for holding on as they looked to run from deep, but a sense of injustice at the disallowed try remained.
Grove produced a huge tackle Hoani Tui but it was Exeter who continued to play a real open style as Arscott this time knocked on after some great handling from Matt Jess and Thomas.
Some more brilliant handling from Exeter around the fringes saw the visitors strike for a second time as they rode tackles before Thomas broke into a gap and fed Steenson on his shoulder who bashed over from close for a 14-6 lead.
Worcester looked to respond with Taulava freeing Matt Mullan on his shoulder, but Arr was caught by his opposite man as he looked to break away and when he was isolated the penalty went against him for holding on.
Warriors were forced to dig really deep to repel more sustained pressure on the home line as Exeter time and again drove before Worcester managed to turnover ball and Goode smashed the ball away to relieved cheers.
An offside decision against the Chiefs, after Jess spilled the ball, gave Goode another kick at goal but his attempt drifted wide left of the uprights.
Lemi looked to spark Worcester into life as he cut inside off his left wing and offloaded to Pennell and Grove as the hosts looked for a score before the break and a way back into the game.
The Samoan speedster again went down the middle and looked to feed Arr but the pass failed to find the hands of the scrum half. Drauniniu was then just inches from picking off a pass as Exeter looked to loop a pass wide before the break.
A high tackle on Chris Jones at the start of the second half saw Goode kick to the corner and Worcester look to assert some early pressure. But the lineout was lost and Exeter cleared away.
A wheeled scrum gave Chiefs a penalty and Steenson a shot at the posts and the in-form fly half made no mistake to increase the lead to 17-6.
Goode produced a cracking kick over Arscott's head, when he had very little on after Matavesi was well tackled, to force Exeter into a lineout deep in their own 22. A penalty from the lineout saw Goode kick for the corner.
The home pack collected and duly drove over to a huge ovation as Worcester sensed a comeback could be on.
Seconds later Pennell broke tackles down the middle and was away before feeding Lemi on his shoulder. But the winger was tackled as he looked to break away and lost the ball as he looked to recycle as the game suddenly got stretched from end to end.
Goode was the next to break the Exeter line and this time he intelligently kicked towards Drauniniu's wing before the penalty was won and the fly half again kicked to the corner.
Again the forwards put the heat on and shunted Exeter back before as it was hauled down the ball was fed to Drauniniu who stepped and dived past his man to score a second half try. Goode added the extras an Worcester led 18-17.
A poor restart allowed Jess to gather for Exeter and when Percival kicked out the lineout went to the Chiefs. Chris Jones, though, picked off the set piece and Goode hammered a long clearance.
Steenson missed a relatively easy penalty to put Exeter back ahead before Arscott latched onto a kick that failed to find touch and raced past men to force the Chiefs deep back in the Worcester half.
Exeter were enjoying their best spell of the second half and as they tore up close the hosts lost a man as Matavesi was yellow carded for deliberately slowing the ball down as Warriors prepared to face the final ten minutes a man down.
The visitors immediately look to strike from close range with the man advantage, but a penalty in the scrum was secured by replacement John Andress that sparked huge home celebrations.
Replacement Sam Betty unloaded a huge hit and clearly wobbly on his legs he was replaced by Taulava.
Exeter looked to strike late for a second successive season at Sixways and drove up close before Dean Mumm broke hearts as he crashed over and Steenson added the extras for a 24-18 lead.
Worcester staged one last attack but when Pennell was bumped into touch the game was over and Warriors had to settle for a losing bonus point.
03m Goode p 3-0
06m Arscott t Steenson c 3-7
13m Goode p 6-7
21m Steenson t Steenson c 6-14
46m Steenson p 6-17
49m Kvesic t 11-17
59m Drauniniu t Goode c 18-17
77m Mumm t Steenson c 18-24
Warriors: 01 Matt Mullan (Ce Jones 73), 02 Ed Shervington (Lutui 73), 03 Euan Murray (Andress 64), 04 James Percival, 05 Chris Jones (Schofield 65), 06 Neil Best, 07 Matt Kvesic, 08 Semisi Taulava (Betty 69, Taulava 74), 09 Jonny Arr, 10 Andy Goode, 11 David Lemi, 12 Josh Matavesi, 13 Alex Grove, 14 Josh Drauniniu, 15 Chris Pennell
Replacements: 16 Aleki Lutui, 17 Ceri Jones, 18 John Andress Murray, 19 Dean Schofield, 20 Sam Betty, 21 Shaun Perry, 22 Danny Gray, 23 Jon Clarke