Morrison struck the opener in the 35th minute at the Britannia Stadium with a 30-yarder that left Stoke goalkeeper Thomas Sorensen red-faced.
Sorensen had brilliantly tipped an earlier Morrison effort onto the crossbar, but on this occasion a last-second swerve on the ball caught out the Dane.
Morrison, who also struck the woodwork twice, then wheeled away in celebration towards the Stoke fans in the Seddon Stand, initially with his arms outstretched.
However, perhaps reacting to a comment, Morrison then quickly placed his left hand on his right upper arm and delivered a fist-pump gesture.
When asked about the incident after the game, Hodgson said: "Did he do that? Blimey. You guys (the media) see everything.
"That gesture could be interpreted in a thousand ways.
"In the Premier League there is so much good you can talk about, so many interesting things to say and do, so to start looking for imaginary gestures, for me that saddens me.
"I hope he didn't do a gesture. If he did, I'm sure he'll apologise for it.
"But he's not that type of guy. I guess it was just a way of showing a celebration, but I didn't see it, and what's more I don't want to."
Hodgson also confirmed captain Jonas Olsson raised no concerns about an incident with Stoke striker Peter Crouch towards the end of the first half.
As the duo clashed, Crouch reached around the front of the centre back, with his right hand coming in contact with Olsson's face.
Olsson did not make a meal of it, with Hodgson adding: "He hasn't said anything.
"I'm afraid with penalty area situations, you could almost prove anything with television replays, someone who has a hold here or who has pushed there.
"But I have to say I thought the referee and his assistants were very good. I thought they officiated fairly and adopted a sensible line.
"They cut out as much of the pushing, shoving and holding as they possibly could. I don't think any referee is capable of seeing everything that goes on."
After Ben Foster saved a 71st-minute penalty from Jonathan Walters, Stoke's second-half substitutes Jermaine Pennant and Cameron Jerome combined for the latter to equalise four minutes from time.
There was still time, however, for one last act as Graeme Dorrans ended a near year-long goal drought by curling home a low 25-yard free-kick in the first minute of injury time, with Sorensen seemingly at fault again.
The win ended Albion's 30-year hoodoo at Stoke having lost 13 and drawn five of the previous 18 matches in the Potteries.
"I'm very pleased with the result, with the performance. It's a very satisfying day," said Hodgson.
"The players who have come in have done a fantastic job, and we end up getting a result against Stoke the club has been waiting for for 30 years."
After one defeat in the previous 10 league matches, Stoke manager Tony Pulis claimed his side came out "flat" for this encounter.
"In the second half we stepped it up a little bit, but the penalty miss is a disappointment," said Pulis.
"I really want to see the two goals conceded because I thought they were desperately disappointing.
"I was as disappointed with the second because anything that is played that side (to Sorensen's left) should be the goalkeeper's.
"If it beats him over the wall, fine, but if it beats you that side...It just wasn't our day."
