Bale beat the offside trap to net in first-half stoppage-time, as Spurs came from behind to beat the Championship's basement club and set-up a fifth-round tie at Everton.
The 31-year-old completed 90 minutes for the first time this season and produced his most influential performance since returning to Spurs on loan, despite betraying his lack of sharpness in front of goal with a succession of misses either side of his goal.
Bale and Hart were both signed in the summer for their personalities and experience as much as their ability, and the goalkeeper says his teammate is proving influential on and off the pitch as Spurs continue to progress in all three cup competitions.
"We always knew it was going to be a project with Gaz and he fully understands that," Hart told Standard Sport. "He's here to help the team win things and he demonstrated that on the pitch last night and on a few other occasions. Gaz's obviously had his injuries but he knows how to look after himself and when he's fit he's going to help our team.
"He's also doing it behind the scenes. His influence is as you'd expect – he's been there and done it. For a club trying to do things, I think it's important to have guys that have been there and won and can reference what it takes. We talk about it a lot but we're a team that wants to win. There's two teams winning a lot – Man City and Liverpool – and there's seven or eight of us trying to win. So we've got to find different ways of doing it."
Spurs manager Jose Mourinho said he never doubted Bale could complete 90 minutes at Adams Park but he can expect to return to the bench for Thursday's visit of faltering champions Liverpool in the League.
Mourinho was relieved to avoid extra-time as Harry Winks scored in the 86th minute to break Wycombe's resolve before substitute Tanguy Ndombele added gloss to the scoreline with a late double. Wycombe, who are rooted to the foot of the Championship table, took a shock first-half lead through Fred Onyedinma.
While Bale demonstrated his enduring quality as a both a poacher and a deep-lying playmaker and Mourinho was delighted with Winks, Spurs lacked a cutting edge until the introductions of Ndombele, Harry Kane and Heung-min Son, further entrenching the feeling that Mourinho is growing reliant on a core group of players.
"Credit to Wycombe, I thought they played really well, made it difficult for us, scored their chance," said Hart. "But I think we played well. They've had a difficult couple of weeks with Covid but our bench, our power and our fitness really shone through."
Spurs lost to a stoppage-time goal at Anfield last month but Hart believes there is little to chose between them and Liverpool ahead of the midweek showdown.
"We feel the same as we did going into that last game," he said. "How the 90 minutes plays out depends on two quality teams against each other. We know we can win it and they know they can win it."