Southgate has taken over as interim manager following the sensational end of Sam Allardyce's 67-day spell at the helm.
The 46-year-old faces qualifiers against Malta and Slovenia next month before matches with Scotland and Spain, the latter a friendly, in November.
Robson feels the former England defender can prove his worth – if he wants the job full-time.
"If Gareth can actually win the next four games before Christmas then there's a great opportunity for Gareth to maybe get the job," said the ex-England captain.
"If that's not the case and Gareth doesn't want the job full-time or the FA feel that you need to move on and try and get somebody with more experience, that's only the FA's decision.
"But I only hope Gareth steps in and he gets the best out of the England squad at this moment in time and we go on to be really dominant, which we have been in the last few qualifying competitions that we've been in."
England claimed a 1-0 win over Slovakia in their first World Cup 2018 qualifier, which was Allardyce's only game in charge.