The Baggies managed to get over the League One side courtesy of a 4-3 penalty shootout win after it ended 1-1 at the end of extra-time at London Road on Wednesday night.
The Posh were good value during the contest and their performance did not go unnoticed by 2004 cup winner Fletcher who scored Albion's equaliser with a cracking half-volley after Jon Taylor's opener.
"That was a real tough match. Their work rate and their energy and the way they played, it was a real cup tie tonight," he told the BBC.
"They came right at us in the first 20 minutes and I thought they were the better team, but once we settled down we had some great chances as well.
"Overall I thought we had the better chances in the game but it was a great cup tie, but in the end someone has to go through and fortunately we've gone through on penalties."
West Brom will now meet Reading at the Madejski Stadium in the fifth round on February 20.