As "New" is a fairly relative term in terms of a religion with a history spanning nearly four thousand years, I will attempt to give an answer by dating the concept of a worship pastor and document the earliest references I was able to locate.
The term "Worship Pastor" appears to be a relatively new term. The Google Books search engine provides a convenient ability to filter based on copyright (publication) date which allows us to search based on year. While Google has indexed several libraries worth of works, it still represents a fraction of English literature ever written. Nonetheless, it is still useful in establishing and trending the rise and fall of certain vernacular in writings and can allow us to establish at least a rough era.
According to this measurement, the actual term "Worship Pastor" does not appear in literature until 1987 in a work titled Exploring Worship: A Practical Guide to Praise & Worship by Bob Sorge. In this Work, Jonathan Stockstill is designated as the Worship Pastor of the Bethany World Prayer Center located in Baton Rouge, Louisiana.
The concept of a worship pastor however, dates much earlier (according to the same measurement). Prior to this timeframe, the worship pastor was typically designated as the Musical Pastor. The concept of music as worship is well established and one need only look at the Psalms to see the connection, so I will not dwell on the point that a music pastor is a kind of worship pastor.
A search on the term "Music Pastor" reveals that this concept dates to the late 1800s with the term first appearing in the 39th volume of Dwight's Journal of Music in 1880.
In an article entitled "Reform of Church Music" attributed to a "Mr. Thayer" he opens by stating
I cannot forego the opportunity of saying a words about organs and organists. Whether professed Christian or not I believe the organist's first duty is to consider his playing and all his acts in the sanctuary as worship.
Mr. Thayer goes on to state
These most briefly stated are the organist's duties and responsibilities; and I believe that he should be fully prepared for them before he assumes the office of musical pastor, or attempts to lead others in the service of the sanctuary.
Still later, he says
The true church organist is a musical pastor who must speak to the hearts of the people, Whoso among us does not feel this, is not yet worthy of his sacred calling.
This tends to indicate that the church organist (who most certainly predated this development by several hundred years, in addition to choral and musical directorship in the church setting) underwent a conversion to a more formal leadership and pastoral role at the end of the 1800s.
Indeed, the second reference I was able to locate of the organist-as-pastor in a paper titled "Church Music" by Albert J Blakesley (the Organist of the Second Congregational Church in Waterbury Connecticut) included with the minutes of the 23rd annual meeting of the General Conference of Congregational churches held at New Haven on June 18th-19th in 1890 supports this conclusion. In this paper, Blakesley states
A church should no more think of having an unchristian man for an organist than such a man for a pastor and teacher. An organist has been rightly described as "the musical pastor" and the term ought to fit every man who sits at the organ desk in divine service... and he must be a good musician, just as good as can be had; but piety before technique and good taste rather than manual dexterity are the qualities to seek.
Taken together, these references indicate that while the office of Musical Pastor did exist in some locales, it was still relatively uncommon and novel idea. Research indicates that this changed relatively shortly thereafter with references being plentiful after turn of the 20th century.