Rangers ended a losing run of five matches, as they clinched a second away victory of the season against 10-man Baggies.
Chris Ramsey’s side are still in the drop zone but are just three points behind fellow strugglers Aston Villa.
Barton, who scored Rangers’ fourth, is eyeing a late run which can take them to safety.
“We took our chances and hopefully it’s a springboard to put a run together,” he said.
Albion are eight points clear of Rangers but suffered their heaviest defeat under Tony Pulis, despite Victor Anichebe’s second-half header.
Myhill, Baird (Olsson 65), McAuley, Lescott, Brunt, Gardner (Mulumbu 72), Fletcher, Morrison, Sessegnon (Anichebe 45), Ideye, Berahino
Green, Isla, Onuoha, Caulker, Yun Suk-young (Hill 82), Phillips, Barton, Sandro, Vargas (Kranjcar 30), Austin, Zamora (Henry 75)