Maintenance for the week of December 23:
• NA megaservers for maintenance – December 23, 4:00AM EST (9:00 UTC) - 9:00AM EST (14:00 UTC)
• EU megaservers for maintenance – December 23, 9:00 UTC (4:00AM EST) - 14:00 UTC (9:00AM EST)
You might just not have seen the complete move - there is a card which is immediately played and allows you to take one other card from the tavern up to the cost of 5 - and there is as well one for up to 7 as well. If that card is played, there is no cost to acquire that chosen card, as long as it doesn't exceed the price limit. If that card was picked up and he played it, he can well have a 5+ card in his 2nd turn.
If something like that happens, I'd love to see a screenshot. It's a lot easier to figure out what's going on that way.
Also, unless they've changed it since PTS, you can click at any time on your opponent's hand and cooldown pile, the tavern and tavern deck and what's been discarded out of the game, as well as your own - it shouldn't be that hard to re-construct what's already gone down on turn 2. (You can't see which cards your opponent has drawn vs. which are still in their deck to draw - those are all mixed together.)
Event Tracker addon (PC NA/EU)
Helps you keep track of your Event Tickets, so you don't miss any. Double XP on events is PASSIVE now!!
If it truly was cheating like you suspect, then I would expect more people to be complaining about it and to have video proof by now. With only you claiming to have seen it, the best guess I have is that something was likely missed.