Honestly there's probably not going to be anything faster than np.inner or np.dot. If you find intermediate variables annoying, you could always create a lambda function:

sqeuclidean = lambda x: np.inner(x, x)

np.inner and np.dot leverage BLAS routines, and will almost certainly be faster than standard elementwise multiplication followed by summation.

In [1]: %%timeit -n 1 -r 100 a, b = np.random.randn(2, 1000000)
((a - b) ** 2).sum()
   ....: 
The slowest run took 36.13 times longer than the fastest. This could mean that an intermediate result is being cached 
1 loops, best of 100: 6.45 ms per loop

In [2]: %%timeit -n 1 -r 100 a, b = np.random.randn(2, 1000000)                                                                                                                                                                              
np.linalg.norm(a - b, ord=2) ** 2
   ....: 
1 loops, best of 100: 2.74 ms per loop

In [3]: %%timeit -n 1 -r 100 a, b = np.random.randn(2, 1000000)
sqeuclidean(a - b)
   ....: 
1 loops, best of 100: 2.64 ms per loop

np.linalg.norm(..., ord=2) uses np.dot internally, and gives very similar performance to using np.inner directly.

For numpy 1.9+

Note that, as perimosocordiae shows, as of NumPy version 1.9, np.linalg.norm(x, axis=1) is the fastest way to compute the L2-norm.

For numpy < 1.9

If you are computing an L2-norm, you could compute it directly (using the axis=-1 argument to sum along rows):

np.sum(np.abs(x)**2,axis=-1)**(1./2)

Lp-norms can be computed similarly of course.

It is considerably faster than np.apply_along_axis, though perhaps not as convenient:

In [48]: %timeit np.apply_along_axis(np.linalg.norm, 1, x)
1000 loops, best of 3: 208 us per loop

In [49]: %timeit np.sum(np.abs(x)**2,axis=-1)**(1./2)
100000 loops, best of 3: 18.3 us per loop

Other ord forms of norm can be computed directly too (with similar speedups):

In [55]: %timeit np.apply_along_axis(lambda row:np.linalg.norm(row,ord=1), 1, x)
1000 loops, best of 3: 203 us per loop

In [54]: %timeit np.sum(abs(x), axis=-1)
100000 loops, best of 3: 10.9 us per loop