The Great Britain centre scored in the first and second periods to secure the win, with Mark Cooper grabbing the game-winner in the first period as netminder Tyler Beskorowany produced a 19-save performance to maintain the Giants' five-point lead at the top of the table.
The Devils started the brighter, although visiting netminder Ben Bowns had to be alert to deny Steve Owre after a nice one-two with captain David Goodwin, while at the other end Beskorowany denied Joey Martin after a giveaway in the Giants end by Will Cullen.
But when the Giants' goals came, they came in quick succession as first Conway produced the one-time finish from the right circle at 11:31 off Owre's feed (1-0) before Cooper got in on the scoring action just 46 seconds later when he finished off a bobbling puck in the slot. (2-0)
Beskorowany needed to be sharp to deny Cole Sanford on the breakaway late in the period, the forward skating in immediately after serving a penalty for boarding but denied by the Giants goaltender.
Cardiff began to ramp up the pressure in the second period, aided by a high-sticking call against Conway, and nearly produced a goal as they had several chances, only to be denied by some excellent blocks in front, including one superb one from Gabe Bast.
The Giants thought they had a third against the run of play when Henrik Eriksson's pass to the crease was caught under the pads of Bowns and Ciaran Long was there to stab it in, only for the officials to wave it off for goaltender interference from the English forward.
But the lead would be three going into the second intermission as Conway doubled his tally for the night, the centre found in the left circle and he was able to battle to retain the puck and find the top corner at 38:40. (3-0)
Looking to add onto their lead, the Giants came close to grabbing a fourth when a snapshot from Owre rang off the post and away, with Bowns producing some stellar saves to keep the visitors in it.
And his forwards rewarded him when they finally got on the board, Trevor Cox going on the breakaway and dishing across for Sanford to provide the final touch to make it a nervy finish at 51:57. (3-1)
However, there was to be no late drama even when the Welsh side pulled Bowns for the extra skater, the Giants able to close up shop around the superb Beskorowany and close out the two-goal win and secure the first two points on offer for the weekend.
The Stena Line Belfast Giants are back on home ice tomorrow evening (Sunday 19 March 2023) for day two of the double header v Cardiff Devils. Tickets on sale HERE.