Griffiths caught Hart out from 25 yards twice in the space of three minutes in the latter stages of Saturday's qualifier.
Griffiths caught Hart out from 25 yards twice in the space of three minutes in the latter stages of Saturday's World Cup qualifier in Glasgow, before Harry Kane's 93rd-minute equaliser rescued a point for the visitors.
Southgate confirmed Hart would sit out Tuesday's friendly against France in Paris, but said that should not be taken as a reflection of the manager's feelings about his goalkeeper's recent displays.
"I think our intention was always to give somebody else a game," Southgate told reporters at Hampden Park.
"Whatever it looks like, I'll have to ride that. You've got to keep making decisions for the right reasons."
Stoke City's Jack Butland, Fraser Forster of Southampton and Burnley's Tom Heaton are in contention to start the game against France.
Hart is in limbo at club level, having spent last season on loan at Italian side Torino after being frozen out at Manchester City by incoming manager Pep Guardiola.
Southgate said it was important for Hart to move to the right club, but said the 30-year-old had made important progress, particularly with regard to criticism that he was getting himself too pumped up for matches.
"I'm sure he'll be playing at a high level. For all of the players, we've competition for places, so we want as many people in every position playing at a high level," Southgate said.
"He's had an important experience going abroad and playing. For me, around the camp, all season his contribution has been really high."
"Without his save in Slovenia (during England's 0-0 draw in October), we'd be worse off in the group as well."
"He's shown great leadership. He's been much more composed before matches. That's been a noticeable shift. For me, two moments of quality (from Griffiths), but we will look at it."
With Kane leading the line in his first match as captain, Marcus Rashford found himself operating in a position wide on the right.
The 19-year-old Manchester United forward has made no secret of his desire to play at centre-forward, but Southgate believes it is not guaranteed that it will turn out to be his best position.
"I don't think it's important at this moment in time that he's pinned down to one position," said Southgate, whose side lead Slovakia by two points at the top of UEFA qualifying Group F.
"He has the ability, and has done right the way through his youth career, to play wide and play central. He's recognised that, talking to him."
"It's a good opportunity for him to learn the game and get a broader view of the game. The reality is, who knows what his best position will be in the end?"
"We all have a feel that it might be as a nine, but maybe not. He is a player that can beat players from wide areas."
"We thought it gave us another goal-scorer coming onto the game. I don't see it as a problem at this stage."