The 25-year-old former Kilmarnock frontman opened his account for the Buddies against Dundee at Dens Park on Saturday in a pulsating game that ended 2-2.
Brophy combined well with strike partner Curtis Main to drill the ball past Dark Blues keeper Adam Legzdins with just four minutes on the clock.
Now, Goodwin is hoping that goal will be the first of many for the forward in this campaign.
He said: "Main and Brophy complement each other really well. You saw that from the first goal.
"Eamonn knows why he is here and what he's paid to do. I won't be putting him under any extra pressure.
"He's a natural goalscorer and you don't need to tell him to take a shot, which I like.
"That finish wasn't an easy one – it's a difficult chance and he's produced a great strike on his weaker left foot.
"It was an unstoppable strike and he will have a very good season. I have no doubt about that."
After that early Brophy goal, Dundee drew level when St Mirren skipper Joe Shaughnessy headed into his own net.
The Buddies took the lead once more when Jamie McGrath scored a hotly-disputed penalty after home striker Jason Cummings was judged by referee Craig Napier to have brought down Jay Henderson.
However, the Paisley men were pegged back again as Cummings added the finishing touch to a sweeping move, with Dundee hanging on to secure a share of the spoils despite being reduced to ten men in the 69th minute after Max Anderson was shown a straight red card.
In Dingwall, Ross County and St Johnstone shared the spoils in a 0-0 draw.
St Johnstone had the best chance to win the match when they were awarded a 70th-minute penalty, but Ali McCann – hitting the spot-kick because regular taker Liam Craig was on the bench – sent his effort over the bar.
Callum Davidson’s Saints now prepare to travel to Turkey ahead of the Europa League third qualifying round tie against Galatasaray in Istanbul.