Warriors' former England fly-half scored 18 points, including a decisive try 12 minutes from time that left Sale still searching for a first league win since April.
The Sharks doubled their Premiership points tally courtesy of a losing bonus after they matched Worcester on tries thanks to touchdowns by wing Mark Cueto and full-back Nick Macleod.
Fly-half Danny Cipriani added two penalties, but Worcester prevailed after Goode's late touchdown followed a scintillating effort early in the action from Warriors' Samoan wing David Lemi.
Goode, the second highest points scorer in Premiership history behind former Sale star Charlie Hodgson, ultimately made the difference with his goalkicking.
The ex-Leicester player slotted three penalties and two conversions for a five from six success-rate, which proved pivotal in denying Sale at least a share of the spoils.
Sale spent the closing minutes within sight of Worcester's line, but they could not capitalise on their forwards' admirable collective effort.
It took a superb try-saving tackle from Worcester scrum-half Paul Hodgson to deny Sale substitute Will Addison in the game's last attack, with both players requiring lengthy treatment after referee Greg Garner blew the final whistle.
Addison limped away, while a similarly groggy Hodgson also returned to his feet, albeit several minutes later.
Worcester, fresh from scoring a staggering 175 points in successive Amlin Challenge Cup victories over Gernika and Rovigo, saw three players - Hodgson, wing Errie Classens and lock Craig Gillies - make their first league starts this term.
Sale also showed numerous changes after their last outing - a 33-18 Heineken Cup defeat against Montpellier - but their Scotland lock Richie Gray failed a late fitness test on his injured ankle and Kearnan Myall deputised.
It took the Warriors barely a minute to open their account as Goode landed a touchline penalty, but Sale's response proved swift and impressive.
From their first attack they retained possession smartly, and Cueto sliced through a gap to score in the right-hand corner, giving him his 74th Premiership try and leaving him just one behind his former team-mate Steve Hanley's all-time league record.
Cipriani missed a difficult conversion attempt, and Worcester quickly regained the lead when Goode slotted an angled penalty to make it 6-5 after nine minutes.
Sale, though, continue to play with a level of confidence and continuity that belied their league status, and it came as no surprise when Cipriani's opening penalty strike restored their advantage.
Worcester struggled noticeably at the lineouts, losing four in succession on their own throw, and they were also under pressure territorially as Sale bossed the close-quarter exchanges.
The visitors had a monopoly on possession as they repeatedly forced Worcester on to the back foot, and Warriors desperately needed some inspiration.
It duly arrived seven minutes before the break thanks to Lemi's brilliance as the Samoan Test skipper jinked his way over on an angled 30-metre run that showcased his devastating finishing ability.
Goode added the conversion before he slotted a penalty on the stroke of half-time as a 10-point onslaught in six minutes gave Warriors a 16-8 interval advantage.
Cipriani opened the second-half scoring with another penalty, and in a repeat of the opening quarter Sale then proceeded to dominate territory.
They camped inside Worcester's 22, using the physical presence of forwards like Andy Powell and Richie Vernon to batter away at Worcester's defence.
But the Warriors held firm and they were able to relieve the pressure when Sale infringed, and Worcester boss Richard Hill then made his first substitution, sending on former Sharks skipper Dean Schofield for Gillies.
Sale, though, were not to be denied their second try, and it arrived in the 62nd minute after more patient approach work this time reaped its reward as Cueto delivered a scoring pass to Macleod.
Cipriani was again wide with the conversion attempt, yet Sale had tied things up at 16-16, ensuring a gripping finale which ultimately belonged to Goode through his converted try that meant the Sharks remain in deep Premiership waters.