The Robins scored two goals in the last 20 minutes to come from 1-0 down, and it looked until they had won the game.
That was right up until the last minute when Morrison tucked home from inside the six-yard box.
Saido Berahino came off the bench and opened the scoring in the 68th minute with his his first goal since October.
He took just four minutes to break the deadlock after head coach Tony Pulis sent him on in a triple substitution.
If the Baggies thought that goal had beaten Steve Cotterill's side who came to The Hawthorns without a win in five, they were wrong.
Seven minutes later, Robins top scorer Jonathan Kodjia nodded in Luke Freeman's free-kick from three yards out.
ALBION (4-4-2): Foster, Chester (Dawson 62), Evans, McAuley, Brunt, McClean (McManaman 62), Morrison, Fletcher, Sessegnon, Lambert (Berahino 62), Rondon.
Unused subs: Myhill, Yacob, Gardner, Gamboa.
Bristol City (3-5-2): Fielding (O'Leary 45), Little, Ayling, Flint, Baker, Bryan, Smith, Pack, Freeman, Wilbraham (Agard 82), Kodjia (Burns 82).
Then in the 83rd minute, Kieran Agard followed up a long-range shot to send the 5,300 City fans into raptures.
Agard had been on the pitch for less than a minute before he grabbed what appeared to be the winner.
But Morrison ensured that the two clubs would do it all again at Ashton Gate in a week and half's time.
Abion had controlled the match for the first 70 minutes but they were lacklustre in the final third and heavily reliant on Stephane Sessegnon.
Before the match, Tony Pulis promised to put out a strong attacking team, and true to his word, the Albion boss filled the 18 with his best players.
He made four changes to the starting line-up, but two of them were bringing James McClean and Salomon Rondon back into the fold after their three-match bans.
Goalkeeper Ben Foster made his first start for the club since damaging his anterior cruciate knee ligament last March and Berahino stayed on the bench for the 10th successive match.
James Chester was handed just his fourth start for the club, since arriving in the summer for £8million.
Right-back Craig Dawson was rested for the first time this season after playing 90 minutes in all previous 22 games.
Aston Villa loanee Nathan Baker recovered from an injury in time to make Bristol City's starting line-up, but the Robins were down to the bare bones and could only name six substitutes.
That didn't dampen the spirits of the travelling fans, who filled out the whole Smethwick End, but Sessegnon nearly did just three minutes in.
Beating Baker to the ball on the half-way line he ran towards the away fans, but after making his way into the box and chopping inside two further defenders, eventually his shot was blocked.
A minute later, McClean robbed Mark Little of the ball on the touchline but Rondon was unable to get a meaningful touch on his pull-back as the Baggies took early control of proceedings.
Pulis had promised end-to-end football and Korey Smith sliced a long-range effort over for the visitors in the ninth minute.
Then Joe Bryan's low curling shot towards the near post called Foster into action for the first time of the afternoon.
Robins goalkeeper Frank Fielding was then forced to beat away McClean's first-time effort from long-range, after Chris Brunt's free-kick hit the wall.
But it was Sessegnon on the other wing who was the spark, and Steve Cotterill's decision to play three at the back was giving the winger too much space.
Even when he was marked, the Benin international had the beating of Bryan the left wing-back and it looked like only a matter of time before one of his jinking runs or balls into the box bore fruit.
But Albion's two strikers have just four goals between them so far this season, and the home side lacked any cutting edge in the box.
Just before half-time, Sessegnon wriggled past Baker on the by-line yet again, but Morrison screwed his shot wide after Rondon had stepped over the pull-back.
Steve Cotterill brought 19-year-old goalkeeper Max O'Leary on in place of Fielding at half-time for his Robins debut.
The teenager was beaten by McClean's header eight minutes into the second half but fortunately for the visitors, the Irish winger's goalbound effort was cleared off the line.
Albion wanted a penalty a minute later when Rondon was barged over in the box, and then shortly after that the Venezuelan somehow pulled his shot wide after Morrison slipped him through.
It was a frenetic start to the second half and Bristol City had their own chances too.
Just before the hour mark, Luke Ayling's deft chip into the box found Kodjia, but although the Robins' top scorer lifted the ball over Foster, his effort hit the side netting.
Determined to force an outcome, Pulis then made a triple substitution, bringing Berahino, Callum McManaman and Dawson on for the McClean and the ineffective Lambert and Chester. They had an immediate impact.
Firstly, McManaman won a free-kick in a dangerous position and then from the resulting attack Berahino opened the scoring.
He squeezed his shot from just inside the box into the bottom right corner. He immediately went over to celebrate with Pulis. The lead only lasted just seven minutes, though.
Kodjia nodded home Luke Freeman's free-kick from the by-line from all of three yards out, giving the Robins fans behind the goal belief they could go on to win the game.
That belief turned into reality ten minutes later when Agard bundled the ball home, after Foster had pulled off a superb strong one-handed save.
It looked for all the world like Cotterill's men had pulled off an upset and they kept the Albion onslaught out nearly long enough, until Morrison levelled things up with almost the last kick of the game.
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