The Premier League leaders were expected to coast past the League Two side at New Meadow in Tuesday's fourth round tie.
However, Chelsea required a late own goal from Jermaine Grandison to avoid extra-time after Andrew Mangan's 77th-minute equaliser had cancelled out Didier Drogba's opener.
Chelsea manager Mourinho opted to rest several first-team regulars such as captain John Terry, Eden Hazard and Cesc Fabregas.
The Portuguese was hoping that the XI chosen against Shrewsbury would cause him a selection headache ahead of Saturday's Premier League clash with QPR, but he was left disappointed.
"I was worried when they equalised. Shrewsbury put up a great fight," he said. "They are what the cups are all about.
"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."
Mourinho did reserve special praise for 36-year-old striker Drogba, though, after he scored for third competitive match in succession and believes Chelsea's young players can learn from his longevity.
"I was worried when they equalised but we go through and Didier played a massive part in that," he added.
"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."
