The 31-year-old midfielder joined the Baggies on a free transfer on deadline day after leaving Manchester United.
Scotland international Fletcher has fallen down the pecking order at United, having been on their books his entire professional career.
Fletcher is in line to make his debut for West Brom on Sunday, when Tony Pulis’s side take on Burnley away from home.
West Brom defender Lescott is delighted to have the Scotsman in the team and is looking forward to playing with him.
“Great to have Darren Fletcher on board,” the 32-year-old wrote on Kicca.com. “Quality, class can't wait to see him in a West Brom shirt this weekend.”
Fletcher will have to hit the ground running at West Brom, who are struggling for survival in the Premier League this season.
The Baggies are currently 15th in the league table with 22 points from 23 matches, just three points above the relegation zone.
West Brom’s Sunday opponents, Burnley, are also one of the top relegation candidates. The Clarets are 17th in the league table at the moment with 20 points, just a point above third-from-bottom Hull City.