The Posh were beaten 4-3 on penalties after it ended 1-1 at the end of extra-time in what was a valiant effort from the League One side against a Premier League opponent.
Peterborough probably had the chances to win the contest in normal time before being bundled out in the dreaded shootout but Westley was not too distraught following his side's brave fourth-round performances which also included a 2-2 draw at The Hawthorns in the initial bout in late January.
"They're an experienced Premier League side and that experience in the end probably told.
"Penalties are a nervy occasion and those that have been there before in big situations came through.
"Our lads are going to gain so much from that experience. They've performed tremendously, played great football and gone toe-to-toe over two games."
The Posh will now focus on their League One campaign as they chase a promotion play-off spot in England's third division.