The points were not safe however until a drop-goal and a penalty deep into injury time from fly-half Chris Malone.
And Bath might have seen victory slip away when forced to contest a five-metre scrum on their own line while they played for the final whistle.
Worcester, whose pack were a constant threat in the close quarters, will be disappointed not to have won even a losing bonus point.
For Bath, showing eight changes from the side that won an epic Powergen Cup semi-final at Gloucester on Sunday, there was only relief that they could finally scrape a win even without 18 first-team squad members.
They even had to recall Ryan Davis to play the entire game despite the young centre having led England Under-21s to victory over Italy the night before.
Worcester fell foul of referee David Rose from the opening minute when tight head prop Chris Horseman conceded a penalty but Malone was off target with the difficult angled kick.
Aided by Worcester's indiscipline – Thinus Delport was sin-binned a couple of minutes later for a late and high tackle – Bath camped in the visitors' half for the next 20 minutes.
Acting captain Zak Feaunati was tackled into touch at the flag and Malone was wide with a drop goal. But they made no mistake with a 35-metre penalty on 11 minutes after Dale Rasmussen's illegal tackle on 19-year-old centre Tom Cheeseman.
Still Worcester would not take the hint about their over-aggressive approach and Horseman followed Delport to the sin bin on 19 minutes after a bust-up at a scrum.
Malone added a second penalty in two minutes, punishing a technical infringement, but Tommy Hayes pulled back three points in the 37th minute as Bath's pack began to wilt under pressure from the Worcester eight.
The visitors might have been level at half-time but Hayes was wide with another kick in stoppage time.
The expected onslaught from the Worcester forwards came immediately after the break with Bath conceding penalty after penalty in the front row and referee Rose eventually awarding a penalty try on 46 minutes, which was converted by Hayes.
Bath's response was to immediately substitute tight head prop Christian Loader with David Flatman and that stopped the rot.
Malone brought Bath back into the game with penalties on 58 and 66 minutes and that slender lead was only just protected before he made the game safe in injury time.
For Bath:Pens: Malone 4Drop: Malone
For Worcester Warriors:Try: PenaltyCon: HayesPen: Hayes