The Hammers found themselves a goal down at half time courtesy of a strike at the far post from Andrei Cordea just after the half-hour mark.
However, Bowen restored parity from the penalty spot with 20 minutes remaining, before goals from Emerson and Antonio ensured the hosts made a winning start in Group B.
Prior to kickoff, a minute's silence was held at the London Stadium to pay respects to Her Majesty The Queen, who has died at the age of 96.
In the opening half-hour of the contest, the Hammers dominated proceedings and boasted over 75% possession, but it was their Romanian visitors who drew first blood in the 34th minute when a low-driven cross from Darius Olaru picked out Cordea unmarked at the far post to slot the ball home.
West Ham responded with three chances from summer arrival Gianluca Scamacca, who firstly glided two headers off target before dragging a right-footed strike well wide after being played through on goal.
Moyes's men were booed off at half time and the Hammers boss decided to ring the changes at the break with Antonio, Lucas Paqueta and Bowen all brought on for Scamacca, Manuel Lanzini and Said Benrahma respectively.
The trio of substitutions provided a much-needed spark to West Ham's frontline, with the physical presence of Antonio, in particular, causing problems to FCSB's back four.
West Ham continued to probe in the final third until they were awarded a penalty midway through the second half, when Maxwel Cornet was taken out by FCSB goalkeeper Stefan Tarnovanu. Cornet was forced off from the resulting challenge and after a lengthy delay, Bowen stepped up to rifle his spot kick high into the roof of the net.
After their equaliser, the Hammers piled on the pressure to FCSB's backline and following a sustained period of pressure, they turned the game on its head in the 75th minute when Emerson found himself in space at the far post before squeezing his effort beyond Tarnovanu into the bottom-left corner, to net his first goal for the club since joining over the summer.
Antonio then wrapped up the victory in the 90th minute when he latched onto a Pablo Fornals pass and surged towards goal before finishing coolly beyond Tarnovanu.
Elsewhere in Group B, Anderlecht secured a narrow 1-0 home win over Silkeborg, with Wolverhampton Wanderers loanee Fabio Silva scoring the only goal from the penalty spot.