But the Premiership side had to battle all the way against League One outfit Morton whose Stefan McCluskey squandered two fine chances.
Morton manager Jim Duffy insisted his side should have had a late penalty and deserved to take the tie to extra time against top flight Thistle.
Duffy agonised as Stefan McCluskey was sent sprawling in a clash with Jags keeper Paul Gallacher in 71 minutes.
He said: "I think it was a penalty and I think it was a poor decision,
"At the end of the day, sometimes you get them and sometimes you don't. It wasn't our night.
"Stefan missed another chance in the first half but overall we competed remarkably well throughout.
"We protected our goal and were always a threat.
"I think it was a bit harsh we didn't take it to extra-time. I am not saying we deserved to win it, but I thought we could've taken it to extra-time.
"It 's all about being in the hat and good luck to Partick Thistle in the next round."
Thistle manager Alan Archibald said of the penalty decision: "Paul (Gallacher) says the boy's leg was trailing and we'll take his word for it!
"I was more angry that a clearance was played over our defence!
"I am just delighted to be through to the next round."