West Brom had gone 21 games without a win – a desperate run stretching all the way back to a Carabao Cup success against Accrington on August 22 – before overcoming Exeter 2-0 in the third round at St James Park on Saturday.
Exeter, like Accrington, might be plying their trade in Sky Bet League Two, but Rodriguez believes cup success can act as a launchpad for Albion to climb out of the relegation zone.
“You see what winning can do for teams in general, no matter what league you’re in,” Rodriguez said.
“When you’re in a winning streak you feel unbeatable and obviously on the other side you find it hard.
“You get into a rut of not winning, you can feel it in the team.
“Those thoughts do creep in, you can’t hide from it that we had not won for a while.”
After Salomon Rondon had struck with just 70 seconds played, Rodriguez’s fifth goal of the season doubled the lead.
The once-capped England forward might have had another as he seemed set to take a 42nd-minute penalty after James McClean had been tripped.
Rondon’s intervention ended up with Hal Robson-Kanu taking, and missing the spot-kick much to the annoyance of West Brom boss Alan Pardew, but Rodriguez insisted Albion’s team spirit is good.
“We’ve got honest lads here who work hard, and I think it’s a really good squad,” he said.
“We know how important the FA Cup is and we want to get on a run in it, but our priority now is to start climbing the Premier League table.
“We’ve got to kick on now, I just feel we need that bit of luck or hang onto a win.”
Exeter almost equalised just before the break when Jayden Stockley struck the post from two yards out and the Grecians produced a spirited second-half performance.
Boss Paul Tisdale reserved special praise for goalkeeper Christy Pym, who he accepts could become a target for rival clubs in the January transfer window.
Two seasons ago, Tom Nichols left Exeter for Peterborough after scoring in an FA Cup third-round tie against Liverpool.
And Tisdale said of Pym: “When you’re that age (22) and you play that well you are going to be in the shop window.
“Christy was excellent, he’s a terrific goalkeeper and his ability to hold the ball is fantastic.”