First order phase transitions in the early universe could produce a gravitational-wave background that might be detectable by the Laser Interferometer Space Antenna (LISA). Such an observation would provide evidence for physics beyond the Standard Model. We study the ability of LISA to observe a gravitational-wave background from phase transitions in the presence of an extragalactic foreground from binary black hole mergers throughout the universe, a galactic foreground from white dwarf binaries, and LISA noise. Modelling the phase transition gravitational wave background as a double broken power law, we use the deviance information criterion as a detection statistic, and Fisher matrix and Markov Chain Monte Carlo methods to assess the measurement accuracy of the parameters of the power spectrum. While estimating all the parameters associated with the gravitational-wave backgrounds, foregrounds, and LISA noise, we find that LISA could detect a gravitational-wave background from phase transitions with a peak frequency of 1 mHz and normalized energy density amplitude of $\OmPeak \simeq 3 \times 10^{-11}$. With $\OmPeak \simeq 10^{-10}$, the signal is detectable if the peak frequency is in the range $4 \times 10^{-4}$ to $9 \times 10^{-3}$ Hz, and the peak amplitude and frequency can be estimated to an accuracy of 10% to 1%.