She was consecrated during a two hour service presided over by the Archbishop of York, Dr John Sentamu, attended by more than 160 people from Cumbria.
Rt Rev Sarah Clark was also consecrated as Bishop of Jarrow in the Diocese of Durham.
Bishop Emma said: "It's been a wonderful day. It's been very special to worship with everyone, listen to a fantastic sermon and hear all those voices singing praise to God.
"'Hills of the North' was one of the hymns that was sung and that seemed very appropriate for Cumbria and Durham!"
The Rt Rev Rachel Treweek, Bishop of Gloucester, who is a long-time friend of Bishop Emma, gave the sermon.
She told the congregation: "It seems a very, very long time ago that Emma first mentioned to me on a campsite at Lee Abbey that she was wondering if she was being called to ordination. Since then it's been wonderful to see how her ordained adventure with God has unfolded among different people and places.
"As Sarah and Emma leave the service today, each with a pastoral staff in their hand, they do so because they are being ordained bishops to be shepherds of Christ's flock, bishops of being 'with': being with God, with people and with place."
Following the consecration she enjoyed a small celebration with family and friends, adding: "I'm very much looking forward to all that is to come in Cumbria."
As part of her new role Bishop Emma will take a lead in the county's 'God for All' project, a plan to see every person in Cumbria have an opportunity to hear about Jesus by 2020.
Since 2015 'God for All' has seen more than 30 new mission communities across the county, with a focus on developing new leaders to support churches, setting up 'fresh expressions' of church and reaching out to people who have no faith.
Commenting after the consecration, Bishop James said: "Personally I found the service hugely moving because it's such a wonderful reminder of what the church is all about and what the bishop's role is, in particular. For Emma this is a whole new chapter of her ministry and a very exciting and significant moment."
Bishop Emma will be officially installed as the Suffragan Bishop of Penrith at a Service of Welcome and Installation at Carlisle Cathedral on Sunday 17 March at 3pm.
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