Your DM has correctly interpreted the rules; but it is possible to cast Eldritch Blast and Cure Wounds in the same turn
The Bonus Action Spellcasting rules, found in the Player's Handbook on page 202, specify that if any spell (cantrip or non-cantrip) is cast as a Bonus Action during a character's turn, then the only spells they are allowed to cast for the rest of that turn are Cantrips that have a casting time of 1 action. So their ruling was correct.
However, Eldritch Blast is a cantrip which requires (normally) an action to cast. If you were to flip it around (Quicken the Cure Wounds and then regularly cast the Eldritch Blast), it would be 100% legal to cast both spells by rules as written. Moreover, there's no requirement in the rules that a Quickened spell must be cast after a regular spell (or vise-versa) so you could still cast these spells in the order you tried to cast them. You just have be more careful about how you apply the metamagic.
I would suggest, out of session, bringing up the rule with your DM (and the rest of your table) and discuss the option of dropping or revising the rule
The confusion you ran into with this rule is confusion that I've run into many times as DM, to the point where I have explicitly Houseruled, at my table, that this rule no longer applies in any 5e games I run. At my table, characters (both PCs and NPCs) are allowed to cast any two spells as Bonus Action and Action, without restriction on the levels of the spells being cast—though, obviously, you still have to legally be able to cast a spell as a Bonus Action, either because that's its normal casting time (like Misty Step) or because you have access to Metamagic/other features that allow Bonus Action spellcasting.
There's three main reasons I made this change. The first is that the D&D 5e designers are on record as saying that the reason they wrote this rule in the first place was to avoid Decision Paralysis at the table—but I have personally found that we actually get the exact opposite problem: players end up getting confused and taking long times to make their turns because the rule is awkwardly written and a little unintuitive, and are constantly asking "okay, I want to cast like this, is this allowed?", especially where Sorcerers are concerned. The Decision Paralysis problem got much better (it happened less often) after we nixed the rule.
The second reason I chose to get rid of this rule is because removing the rule offers a very substantial power boost to Sorcerers, a class that I think is somewhat undercooked in the 5e rules. And I can confirm that at my table, even with substantial Homebrew to boost their Sorcerer features and this rule removed on a character that did take the Quicken Spell metamagic, the sorcerer ultimately ended being pretty average on the scale of power within the party. So the fact that removing the rule boosts Sorcerers [who choose to take the Quicken Spell Metamagic] substantially is not something I see as a problem.
And the final reason is that there's a very unintuitive interaction with this rule, which is that you also lose the ability to cast spells as Reactions during the turn. It doesn't come up often, but you do end up with Counterspell Wars (i.e. an NPC tries to Counterspell the PC's spell, and they try to Counterspell in return), and if this rule is in place, then the PC is unable to use Counterspell if they've cast a Bonus Action spell. Because, again, the rule states that the only other spells that may be cast are Cantrips with a casting time of 1 action. This situation can be a little marginal, but when it does come up, the interaction with this rule has always been frustrating to my players.
So for those reasons, I would suggest you float the idea of nixing the Bonus Action Spellcasting rule to your DM. I find it to be cumbersome to adjudicate at my table, find it really doesn't offer value for its intended purpose, and I feel the way that its removal affects game balance is acceptable.