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A recent emerging trend in the literature on learning in games has been concerned with providing faster learning dynamics for correlated and coarse correlated equilibria in normal-form games. Much less is known about the significantly more challenging setting of extensive-form games, which can capture both sequential and simultaneous moves, as well as imperfect information. In this paper we establish faster no-regret learning dynamics for \textit{extensive-form correlated equilibria (EFCE)} in multiplayer general-sum imperfect-information extensive-form games. When all players follow our accelerated dynamics, the correlated distribution of play is an $O(T^{-3/4})$-approximate EFCE, where the $O(\cdot)$ notation suppresses parameters polynomial in the description of the game. This significantly improves over the best prior rate of $O(T^{-1/2})$. To achieve this, we develop a framework for performing accelerated \emph{Phi-regret minimization} via predictions. One of our key technical contributions -- that enables us to employ our generic template -- is to characterize the stability of fixed points associated with \emph{trigger deviation functions} through a refined perturbation analysis of a structured Markov chain. Furthermore, for the simpler solution concept of extensive-form \emph{coarse} correlated equilibrium (EFCCE) we give a new succinct closed-form characterization of the associated fixed points, bypassing the expensive computation of stationary distributions required for EFCE. Our results place EFCCE closer to \emph{normal-form coarse correlated equilibria} in terms of the per-iteration complexity, although the former prescribes a much more compelling notion of correlation. Finally, experiments conducted on standard benchmarks corroborate our theoretical findings.