The 30-year-old headed past Baggies goalkeeper Boaz Myhill in the 75th minute to set up a fourth-round encounter against the MK Dons.
It was Bothroyd’s first goal of the season for QPR after returning from a loan spell at Sheffied Wednesday in January.
“Jay Bothroyd did well,” Redknapp told BBC Sport. “He was written off. He’s been at Sheffield Wednesday and not done very well.
“They sent him back and I thought I’d give him a go tonight and see what he could do.
“He did well but then again, he played for England a couple of years ago under Fabio Capello.”
Redknapp also praised another QPR player with a point to prove, Rob Green, after the England goalkeeper made a series of saves to deny Romelu Lukaku.
“It was a tough match,” he added. “The first half was a bit dour, not a lot happening but Rob Green made a fantastic save in the first half (from Romelu Lukaku’s header).
“Then when he came under pressure in the second half, he made another couple of good saves after we scored.”
For West Brom boss Steve Clarke, it was yet another setback after such a promising start to the season, but the Scotsman hopes his side can get back on track with a victory over struggling Aston Villa at the weekend.
“I was disappointed, I thought we had the better chances and Rob Green made some good saves but we went to sleep at a set play,” said Clarke.
“The squad’s been stretched in recent weeks, the boys have done their best. Maybe the luck we had a few weeks ago has deserted us.
“But as a group, we know what we are trying to do. There’s no better team to try and turn it around than against Aston Villa at the weekend.”