Gas giant exoplanets orbiting close to their parent stars (``hot Jupiters") experience radiation and stellar wind flux $\sim 10^4$ times higher than solar system giants. Energy deposited at high altitude heats and ionizes their upper atmospheres, where densities are sufficiently low that magnetic forces can dominate the dynamics of the gas --- physics that previous studies of their upper atmospheres have largely ignored. High levels of extreme-ultraviolet radiation deposited into the upper atmosphere inflates the scale height, making the upper atmosphere of hot Jupiters that transit the disk of their host stars potentially observable through transmission spectroscopy of atomic resonance lines. Motivated by the $\simeq 10\%$ decrease in hydrogen (H) Lyman $\alpha$ flux observed for the hot Jupiter HD 209458b, and the interpretations in the literature that the absorbing neutral H gas lies outside the planet's Roche Lobe and may be escaping, I perform semi-analytic calculations and 2D magnetohydrodynamical (MHD) simulations of photoionization-driven escape of gas from the planet. The high ionization levels expected in the upper atmosphere imply that any outflow would be well-coupled to the planetary magnetic field. I have constructed the first models of the upper atmosphere that include the effects of the intrinsic planetary magnetic field and the stellar tide. The solutions exhibit the following three features: (1) a region near the equator of static, magnetically-confined gas, (2) a transonic outflow at mid-latitudes in a magnetically-channeled wind zone, and (3) a region near the poles where outflow can be quenched by a sufficiently strong stellar tide. Using the magnetized wind model, I compute Lyman $\alpha$ transit profiles using several different simulation parameters, to compare with available observational data for the hot Jupiters HD 209458b and HD 189733b. I also use the consistency with observations to offer an alternative to the simpler, hydrodynamic escape interpretation for extended H absorption seen in the transmission spectra of highly irradiated gas giants such as HD 209458b. The results demonstrate (1) the importance of magnetic forces and stellar tidal forces for an accurate determination of mass and angular momentum loss rates, and (2) absorption in the Lyman $\alpha$ line at $\pm 100 $km s$^{-1}$ from line center can occur from regions outside the planet's Roche Lobe without requiring mass loss to occur at all latitudes. Mass and angular momentum loss rates, which are not directly accessible through observations, determine if significant atmospheric ``evaporation" and/or deviation from tidal synchronization occurs on Gyr timescales. The utility of a model for magnetized upper atmospheres of hot Jupiters can be extended to additional classes of exoplanets, such as hot Neptunes, which are Neptune-sized planets with tight orbits around their parent stars, and perhaps even super Earths ($M_{\earth} \lesssim M \lesssim 10 M_{\earth}$). A future refinement of the model is to include the stellar wind contribution, which sets the outer boundary for planetary atmospheres.