The striker fired in his third goal in as many games as the Blues edged into the Capital One Cup quarter-final with a 2-1 win at Greenhous Meadow.
Jermaine Grandison's late own goal won it following Drogba's opener, which followed the Ivorian's strikes against Maribor and Manchester United in the last week.
Diego Costa is expected to return against QPR on Saturday following a virus and hamstring and groin problems.
But boss Mourinho said Drogba was an example to his squad after his key impact.
He said: "Yes I was worried when they equalised but we go through and Didier played a massive part in that.
"Hopefully he will be good for the weekend but we will have to see because right now his character is stronger than his body.
"Maybe what he has done today is through his character - not his body.
"That's what makes players special and any of the young players lucky enough to play alongside him, learn from that."
Andy Mangan's 77th-minute equaliser - just 84 seconds after he came on - rocked Chelsea before Grandison's gaffe when he turned in Willian's cross settled the game for the Barclays Premier League leaders.
Mourinho admitted he feared extra-time, though.
"Yes I was worried when they equalised
Shrewsbury put up a great fight
They are what the cups are all about," he said
"I expect players to give me problems
I love problems
But a lot of them didn't and they've made it easy to choose my team for Saturday."
Drogba added: "I played here before, with Galatasaray
Shrewsbury have improved since then
It was difficult with the weather
I wouldn't say I enjoyed the game
It was difficult but I started playing in places like this.''
Shrewsbury scared Chelsea and impressed in the first half with Nathaniel Knight-Percival forcing Petr Cech into a save and Bobby Grant went close.
They return to Sky Bet League Two action at Dagenham on Saturday and boss Micky Mellon wants to build on the performance.
"Maybe we could have held on for a wee bit longer, we had possession and Mickey (Demetriou) slips and we get done on the counter attack," he said.
"It's all about focus now and concentrating on what we believe
We won't get carried away and we know our bread and butter comes on Saturday.
"A lot of people would have watched that and we'll become a bit of a scalp now
We'll deal with that.
"We haven't done anything yet other than perform well and get knocked out of the cup by a top side
It's important we keep moving forward because the club was on its backside when we came in."
Ryan Woods excelled in the middle for Town and Mellon admits he is determined to keep hold of the home grown 20-year-old.
He said: "He is special and I'm desperate to hold on to him
He is the first player on the training pitch and the last off
He's a terrific kid and has some unbelievable ability."
Source : PA