There is a simple non-analytic proof for $p\equiv 1 \bmod n$; see e.g. Proposition $3$ in [this note][1].  The proof gives a (Euclidean) argument that infinitely many primes divide the values of an integer-coefficient polynomial on the integers, and then notes that the prime divisors of the values of the $n$-th cyclotomic polynomial either divide $n$ or have remainder $1$ upon division by $n$.  (The proof is well-known; I don't know the originator.)

Edit:  [This paper][2] by Murty and Thain discusses obstructions to Euclid-style proofs for various congruence classes.  I believe that a proof has been carried out for $p\equiv a\bmod b$ for $(a, b)=1$ for all $b\leq 24$ in the style of Euclid, however.

Here is an open-access [paper][3] by Keith Conrad expositing this impossibility theorem and giving some background.

Edit 2:  By the way, the paper also contains a cute analytic argument for $p\equiv 1 \bmod 4$ giving bounds on the partial sums of the reciprocals of such primes; the argument uses representations via sums of two squares.

Edit 3:  Here is the [paper][4] I recalled with the Euclidean proof for $b\leq 24$; unfortunately it is not open-access.  It is JSTOR however so many of you likely have institutional access.


  [1]: http://stanford.edu/~dalitt/primes1mod4.pdf
  [2]: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.facm/1229442627&page=record
  [3]: http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/dirichleteuclid.pdf
  [4]: http://www.jstor.org/pss/2310975