Leicester produced a brave fightback but in a frantic finale the home side held on to spark wild celebrations from home fans.
Fan's favourite Thinus Delport led the team out to a huge ovation on his 100th Premiership start as Warriors looked to build on the morale-boosting win over Sale Sharks in the last outing.
But it went wrong almost immediately as Warriors fell behind after only 21 seconds when Drahm had a kick charged down by the onrushing forwards and the ball ran kindly into the try area for Brett Deacon to crash on the loose ball. Goode added the extras for a 0-7 within a minute of the kick-off.
Worcester looked to respond from the shellshock with a penalty giving Drahm the chance to register the first points of the afternoon for the home side. The fly half connected brilliantly to slot a kick from just inside the Leicester half.
Tigers ran the ball back with real intent from the restart and a high tackle from Pat Sanderson on Andy Goode gave the visitors a chance to edge further ahead. The recalled England fly half hit his kick with a real hook and saw it crash against the top of the upright before Drahm booted the ball clear.
Drahm was showing all the signs of his display at Sale and slotted another penalty with real ice cool accuracy to cut the lead to a single point and cap a real high tempo start to the clash.
Leicester kicked for the corner after Worcester were penalised in the scrum and took the ball on through the forwards only to hit and gold and blue wall of defiance as the home side stood firm and won a penalty back.
Worcester looked to spring an attack wide right though Sam Tuiutpou and Delport but the final bal to Marcel Garvey slipped into touch instead of finding the hands of the flying wing.
Both teams were traded some excellent kicks as they looked to gain the territorial advantage in an engrossing opening to the game with Tigers the first to crack when wing Alesana Tuilagi knocked on after a spitting grubber kick from Matt Powell.
Ollie Smith looked to take advantage of a Drahm slip but his flick pass tor Tom Varndell drifted forward and into touch before Warriors roared back with Aleki Lutui making yards and the team showing great attacking intent after an initial break from Garvey.
Worcester were showing some stunning handling as they ran the ball left and right with real slick passing before Tigers were hit for hands in a ruck. However, Drahm saw his penalty drift wide right as the strong wind continued to whip around Sixways.
Drahm looked to make amends with an ambitious cross field kick to Craig Gillies but the ball flew out on the full instead of finding the giant lock.
Leicester looked to have made the second try of the afternoon when Goode made a break in midfield and fed Jordan Crane to take the ball up close. Tigers pounded on the Worcester defence but could find no way through and settled for a penalty after the hosts strayed offside.
Crane again caused damage from the restart as he made yards but Leicester gave the ball away and Garvey returned it with a stunning break only for Tom Wood's pass wide right to fail to find Drahm as numbers rolled up on the wing.
Worcester then stole a Leicester lineout through Gilles and Drahm put them deep in their own half with a crunching kick to the corner as Warriors looked to keep the high flying visitors at bay.
Warriors struck just three minutes from half-time when a flowing move saw them go right with real pace and big lock Greg Rawlinson showed all his power to bulldoze over for a stunning score and a one-point lead.
But just seconds from the break a penalty to Tigers gave Goode the chance to again put his side ahead and he made no mistake.
Warriors came out pumped up for the second half and scored a sensational try after three minutes when Drahm cut a superb line and fed Garvey who tore away from Tuilagi to score. Drahm added the extras for an 18-13 lead.
Tuilagi looked to make amends for missing his man from the restart with a big run only to be bundled into touch by the Worcester forward pack eager for the battle.
Worcester were again on the attack and a stunning Drahm run from deep and kick ahead saw Tigers sliced apart. The ball popped up to Rasmussen who went close before neat hands put in Benjamin for a quite breathtaking score with the home support in absolute raptures.
Warriors were suddenly winning a succession of penalties and had Tigers rocked after a quite staggering ten minute opening spell to the second half.
The rain started to lash down at the home of the Warriors but the home side continued to try and play running rugby all over the field before Wood was penalised at the breakdown. Goode lotted to cut the lead to seven points.
Goode was lucky to escape when a hack forward from Drahm nearly spilled out to Benjamin before Garvey made another break but his inside pass fell forward as both teams continued to slug it out.
Drew Hickey was introduced for Wood as Leicester started to turn the screw as they looked for a try that would put them back on level terms.
Warriors were now defending for their lives with a series of big tackles going in to try and keep Tigers at arm's length. An offside decision against Worcester saw Goode again slice the deficit to four points with just over ten minutes remaining.
Will Bowley and Matt Mullan entered the field of battle but now it was increasingly Tigers who were in the ascendancy with the wind at their backs and Goode hitting monster clearances.
Warriors were now defending with everything they had to keep the visitors away from their line with five minutes left and tension growing.
Leicester threw everything at Warriors in the last few seconds but the forwards won back the ball before Tuitupou ran into touch to spark wild celebrations from the delighted home support.
1m B Deacon t Goode c 0-7
4m Drahm p 3-7
10m Drahm p 6-7
28m Goode p 6-10
37m Rawlinson t 11-10
40m Goode p 11-13
43m Garvey t Drahm c 18-13
47m Benjamin t 23-13
57m Goode p 23-16
67m Goode p 23-19
Warriors: 01 Tony Windo (Mullan 70), 02 Aleki Lutui, 03 Tevita Taumoepeau, 04 Greg Rawlinson (Bowley 70), 05 Craig Gillies, 06 Tom Wood (Hickey 63), 07 Pat Sanderson, 08 Kai Horstmann, 09 Matt Powell (R Powell 68), 10 Shane Drahm, 11 Marcel Garvey, 12 Sam Tuitupou, 13 Dale Rasmussen, 14 Miles Benjamin, 15 Thinus Delport
Replacements: 16 Matt Mullan, 17 Shaun Ruwers, 18 Will Bowley, 19 Drew Hickey, 20 Ryan Powell, 21 Joe Carlisle, 22 Chris Pennell