Just like he has at Spurs all season, Andros took the game to the opposition and was rewarded with a superb debut goal in the 78th a minute, a fierce drive with his right foot that flew in off the post.
That made it 3-1 at Wembley and calmed any fears after Montenegro got themselves back into the game at 2-1 after 71 minutes.
Earlier, Andros' burst down the right set-up England's opener for Wayne Rooney on 48 minutes.
An own goal doubled the lead, Dejan Damjanovic responded but then Andros' 25-yarder and Daniel Sturridge's late penalty saw the Three Lions home 4-1.
England now know that victory in the final Group H match against Poland at Wembley on Tuesday night will see them qualify automatically for Brazil, 2014.
Hodgson said: "He (Andros) made a big impact. He got the man of the match which he must be delighted with. It can't be easy for a young man making his debut in a game of that nature.
"He deserves all the credit which I am sure he is going to get because he played well, with a calmness and composure which I have seen him play with for QPR and most recently Tottenham. I was fortunate he was able to reproduce it.
"We thought Townsend's pace against a packed defence would open doors. He really performed on the night."